fsuppind - Mathbox for Steven Nguyen

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Theorem fsuppind 42576

Description: Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 21793 and frlmplusgval 21801). This theorem is structurally general for polynomial proof usage (see mplelbas 22028 and mpladd 22046). Note that hypothesis 0 is redundant when 𝐼 is nonempty. (Contributed by SN, 18-May-2024.)

**Hypotheses**
Ref	Expression
fsuppind.b	⊢ 𝐵 = (Base‘𝐺)
fsuppind.z	⊢ 0 = (0_g‘𝐺)
fsuppind.p	⊢ + = (+_g‘𝐺)
fsuppind.g	⊢ (𝜑 → 𝐺 ∈ Grp)
fsuppind.v	⊢ (𝜑 → 𝐼 ∈ 𝑉)
fsuppind.0	⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
fsuppind.1	⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
fsuppind.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘_f + 𝑦) ∈ 𝐻)

**Assertion**
Ref	Expression
fsuppind	⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)

Distinct variable groups: 𝑥, + ,𝑦 0 ,𝑎,𝑏,𝑥 𝑦, 0 𝐼,𝑎,𝑏,𝑥 𝑦,𝐼 𝐻,𝑏 𝑦,𝐻,𝑥 𝐻,𝑎 𝜑,𝑥,𝑦 𝜑,𝑎,𝑏 𝐵,𝑎,𝑏,𝑥 Allowed substitution hints: 𝐵(𝑦) + (𝑎,𝑏) 𝐺(𝑥,𝑦,𝑎,𝑏) 𝑉(𝑥,𝑦,𝑎,𝑏) 𝑋(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem fsuppind Dummy variables 𝑧 𝑐 ℎ 𝑚 𝑣 𝑖 𝑗 𝑛 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsuppind.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐺)
2	1	fvexi 6920	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ V)
4		fsuppind.v	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	3, 4	elmapd 8878	. . . . . . . 8 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ↑_m 𝐼) ↔ 𝑋:𝐼⟶𝐵))
6	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵 ↑_m 𝐼) ↔ 𝑋:𝐼⟶𝐵))
7		eqeq1 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 1 → (𝑖 = (♯‘(ℎ supp 0 )) ↔ 1 = (♯‘(ℎ supp 0 ))))
8	7	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 1 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
9	8	ralbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 1 → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
10		eqeq1 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑗 → (𝑖 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(ℎ supp 0 ))))
11	10	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑗 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
12	11	ralbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
13		eqeq1 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = (𝑗 + 1) → (𝑖 = (♯‘(ℎ supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
14	13	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = (𝑗 + 1) → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
15	14	ralbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑖 = (𝑗 + 1) → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
16		eqeq1 2738	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑛 → (𝑖 = (♯‘(ℎ supp 0 )) ↔ 𝑛 = (♯‘(ℎ supp 0 ))))
17	16	imbi1d 341	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑛 → ((𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
18	17	ralbidv 3175	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑛 → (∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑖 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
19		eqcom 2741	. . . . . . . . . . . . . . . . 17 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ (♯‘(ℎ supp 0 )) = 1)
20		ovex 7463	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ supp 0 ) ∈ V
21		euhash1 14455	. . . . . . . . . . . . . . . . . 18 ⊢ ((ℎ supp 0 ) ∈ V → ((♯‘(ℎ supp 0 )) = 1 ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 )))
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘(ℎ supp 0 )) = 1 ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 ))
23	19, 22	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ ∃!𝑐 𝑐 ∈ (ℎ supp 0 ))
24		elmapfn 8903	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (ℎ ∈ (𝐵 ↑_m 𝐼) → ℎ Fn 𝐼)
25	24	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → ℎ Fn 𝐼)
26	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 𝐼 ∈ 𝑉)
27		fsuppind.z	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 0 = (0_g‘𝐺)
28	27	fvexi 6920	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 0 ∈ V)
30		elsuppfn 8193	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
31	25, 26, 29, 30	syl3anc 1370	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
32	31	eubidv 2583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
33		df-reu 3378	. . . . . . . . . . . . . . . . . 18 ⊢ (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 ))
34	32, 33	bitr4di 289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ))
35	24	ad2antlr 727	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ Fn 𝐼)
36		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ‘𝑥) ∈ V
37	36, 28	ifex 4580	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ) ∈ V
38		eqid 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))
39	37, 38	fnmpti 6711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) Fn 𝐼
40	39	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) Fn 𝐼)
41		eqeq1 2738	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ↔ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
42		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (ℎ‘𝑥) = (ℎ‘𝑣))
43	41, 42	ifbieq1d 4554	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 = 𝑣 → if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ) = if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ))
44	43, 38, 37	fvmpt3i 7020	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))‘𝑣) = if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ))
45	44	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))‘𝑣) = if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ))
46		eqidd 2735	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) ∧ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → (ℎ‘𝑣) = (ℎ‘𝑣))
47		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
48		simplr 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )
49		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑣 → (ℎ‘𝑐) = (ℎ‘𝑣))
50	49	neeq1d 2997	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑣 → ((ℎ‘𝑐) ≠ 0 ↔ (ℎ‘𝑣) ≠ 0 ))
51	50	riota2 7412	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑣 ∈ 𝐼 ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ((ℎ‘𝑣) ≠ 0 ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣))
52	47, 48, 51	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((ℎ‘𝑣) ≠ 0 ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣))
53		necom 2991	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ( 0 ≠ (ℎ‘𝑣) ↔ (ℎ‘𝑣) ≠ 0 )
54		eqcom 2741	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ↔ (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) = 𝑣)
55	52, 53, 54	3bitr4g 314	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ( 0 ≠ (ℎ‘𝑣) ↔ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
56	55	biimpd 229	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ( 0 ≠ (ℎ‘𝑣) → 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
57	56	necon1bd 2955	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (¬ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → 0 = (ℎ‘𝑣)))
58	57	imp 406	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) ∧ ¬ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → 0 = (ℎ‘𝑣))
59	46, 58	ifeqda 4566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → if(𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑣), 0 ) = (ℎ‘𝑣))
60	45, 59	eqtr2d 2775	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (ℎ‘𝑣) = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))‘𝑣))
61	35, 40, 60	eqfnfvd 7053	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )))
62		riotacl 7404	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 → (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼)
63	62	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼)
64		elmapi 8887	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ ∈ (𝐵 ↑_m 𝐼) → ℎ:𝐼⟶𝐵)
65	64	ad2antlr 727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ:𝐼⟶𝐵)
66	65, 63	ffvelcdmd 7104	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∈ 𝐵)
67		fsuppind.1	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
68	67	ralrimivva 3199	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
69	68	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
70		eqeq2 2746	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 = 𝑎 ↔ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
71	70	ifbid 4553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ))
72	71	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )))
73	72	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )) ∈ 𝐻))
74		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (ℎ‘𝑥) = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
75	74	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑏 = (ℎ‘𝑥) ↔ 𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ))))
76	75	biimparc 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∧ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → 𝑏 = (ℎ‘𝑥))
77	76	ifeq1da 4561	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ))
78	77	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )))
79	78	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) ∈ 𝐻))
80	73, 79	rspc2va 3633	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∈ 𝐼 ∧ (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )) ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) ∈ 𝐻)
81	63, 66, 69, 80	syl21anc 838	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 )) ∈ 𝐻)
82	61, 81	eqeltrd 2838	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → ℎ ∈ 𝐻)
83	82	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 → ℎ ∈ 𝐻))
84	34, 83	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) → ℎ ∈ 𝐻))
85	23, 84	biimtrid 242	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
86	85	ralrimiva 3143	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
87		fvoveq1 7453	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (♯‘(𝑚 supp 0 )) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
88	87	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑗 = (♯‘(𝑚 supp 0 )) ↔ 𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ))))
89		oveq1 7437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
90	89	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → (𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) ↔ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
91	88, 90	anbi12d 632	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) ↔ (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))))
92		fsuppind.g	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝐺 ∈ Grp)
93	1, 27	grpidcl 18995	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵)
94	92, 93	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ∈ 𝐵)
95	94	ad5antr 734	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → 0 ∈ 𝐵)
96		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ↑_m 𝐼) = (𝐵 ↑_m 𝐼)
97		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
98	97	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
99		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
100	96, 98, 99	mapfvd 8917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → (𝑙‘𝑥) ∈ 𝐵)
101	95, 100	ifcld 4576	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑥 ∈ 𝐼) → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ 𝐵)
102	101	fmpttd 7134	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))):𝐼⟶𝐵)
103	2	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐵 ∈ V)
104	4	ad4antr 732	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐼 ∈ 𝑉)
105	103, 104	elmapd 8878	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∈ (𝐵 ↑_m 𝐼) ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))):𝐼⟶𝐵))
106	102, 105	mpbird 257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∈ (𝐵 ↑_m 𝐼))
107	106	adantrl 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∈ (𝐵 ↑_m 𝐼))
108		ovexd 7465	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙 supp 0 ) ∈ V)
109		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ 𝐼)
110		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙‘𝑧) ≠ 0 )
111		elmapfn 8903	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑙 ∈ (𝐵 ↑_m 𝐼) → 𝑙 Fn 𝐼)
112	111	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
113	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑙 Fn 𝐼)
114	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝐼 ∈ 𝑉)
115	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 0 ∈ V)
116		elsuppfn 8193	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
117	113, 114, 115, 116	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
118	109, 110, 117	mpbir2and 713	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ (𝑙 supp 0 ))
119		simpllr 776	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ)
120	119	nnnn0d 12584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ₀)
121		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑗 + 1) = (♯‘(𝑙 supp 0 )))
122	121	eqcomd 2740	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘(𝑙 supp 0 )) = (𝑗 + 1))
123		hashdifsnp1 14541	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ₀) → ((♯‘(𝑙 supp 0 )) = (𝑗 + 1) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗))
124	123	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑙 supp 0 ) ∈ V ∧ 𝑧 ∈ (𝑙 supp 0 ) ∧ 𝑗 ∈ ℕ₀) ∧ (♯‘(𝑙 supp 0 )) = (𝑗 + 1)) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
125	108, 118, 120, 122, 124	syl31anc 1372	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = 𝑗)
126		eldifsn 4790	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧))
127		fvex 6919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙‘𝑥) ∈ V
128	28, 127	ifex 4580	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ V
129		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))
130	128, 129	fnmpti 6711	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼
131	130	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼)
132	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝐼 ∈ 𝑉)
133	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 0 ∈ V)
134		elsuppfn 8193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 )))
135	131, 132, 133, 134	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 )))
136		iftrue 4536	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 )
137		olc 868	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
138	136, 137	2thd 265	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
139		iffalse 4539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = (𝑙‘𝑣))
140	139	eqeq1d 2736	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ (𝑙‘𝑣) = 0 ))
141		biorf 936	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 )))
142		orcom 870	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧) ↔ (𝑣 = 𝑧 ∨ (𝑙‘𝑣) = 0 ))
143	141, 142	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (¬ 𝑣 = 𝑧 → ((𝑙‘𝑣) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
144	140, 143	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (¬ 𝑣 = 𝑧 → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
145	138, 144	pm2.61i 182	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
146	145	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = 0 ↔ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
147	146	necon3abid 2974	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ↔ ¬ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧)))
148		neanior 3032	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧) ↔ ¬ ((𝑙‘𝑣) = 0 ∨ 𝑣 = 𝑧))
149	147, 148	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ↔ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧)))
150	149	anbi2d 630	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧))))
151		anass 468	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧) ↔ (𝑣 ∈ 𝐼 ∧ ((𝑙‘𝑣) ≠ 0 ∧ 𝑣 ≠ 𝑧)))
152	150, 151	bitr4di 289	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ) ↔ ((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧)))
153		equequ1 2021	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑧 ↔ 𝑣 = 𝑧))
154		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑙‘𝑥) = (𝑙‘𝑣))
155	153, 154	ifbieq2d 4556	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
156	155, 129, 128	fvmpt3i 7020	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
157	156	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
158	157	neeq1d 2997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 ↔ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 ))
159	158	pm5.32da 579	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 ) ↔ (𝑣 ∈ 𝐼 ∧ if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) ≠ 0 )))
160	112	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝑙 Fn 𝐼)
161		elsuppfn 8193	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
162	160, 132, 133, 161	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
163	162	anbi1d 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧) ↔ ((𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 ) ∧ 𝑣 ≠ 𝑧)))
164	152, 159, 163	3bitr4d 311	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ 𝐼 ∧ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))‘𝑣) ≠ 0 ) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧)))
165	135, 164	bitr2d 280	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧) ↔ 𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
166	126, 165	bitrid 283	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ 𝑣 ∈ ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
167	166	eqrdv 2732	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑙 supp 0 ) ∖ {𝑧}) = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 ))
168	167	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
169	168	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (♯‘((𝑙 supp 0 ) ∖ {𝑧})) = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
170	125, 169	eqtr3d 2776	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )))
171	127, 28	ifex 4580	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) ∈ V
172		eqid 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
173	171, 172	fnmpti 6711	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) Fn 𝐼
174	173	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) Fn 𝐼)
175		inidm 4234	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐼 ∩ 𝐼) = 𝐼
176	131, 174, 132, 132, 175	offn 7709	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) Fn 𝐼)
177	153, 154	ifbieq1d 4554	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
178	177, 172, 171	fvmpt3i 7020	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ 𝐼 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
179	178	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
180	131, 174, 132, 132, 175, 157, 179	ofval 7707	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))‘𝑣) = (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )))
181	92	ad4antr 732	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝐺 ∈ Grp)
182		simplrl 777	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ ((𝑙‘𝑧) ≠ 0 ∧ 𝑣 ∈ 𝐼)) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
183	182	anassrs 467	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
184		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → 𝑣 ∈ 𝐼)
185	96, 183, 184	mapfvd 8917	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (𝑙‘𝑣) ∈ 𝐵)
186		fsuppind.p	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ + = (+_g‘𝐺)
187	1, 186, 27	grplid 18997	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ( 0 + (𝑙‘𝑣)) = (𝑙‘𝑣))
188	1, 186, 27	grprid 18998	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ((𝑙‘𝑣) + 0 ) = (𝑙‘𝑣))
189	187, 188	ifeq12d 4551	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)))
190	181, 185, 189	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 )) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)))
191		ovif12 7532	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 ))
192		ifid 4570	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)) = (𝑙‘𝑣)
193	192	eqcomi 2743	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣))
194	190, 191, 193	3eqtr4g 2799	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = (𝑙‘𝑣))
195	180, 194	eqtr2d 2775	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → (𝑙‘𝑣) = (((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))‘𝑣))
196	160, 176, 195	eqfnfvd 7053	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑙‘𝑧) ≠ 0 ) → 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
197	196	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
198	170, 197	jca 511	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
199	198	adantllr 719	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑗 = (♯‘((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) supp 0 )) ∧ 𝑙 = ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
200	91, 107, 199	rspcedvdw 3624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → ∃𝑚 ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
201	111	ad2antrl 728	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 Fn 𝐼)
202	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝐼 ∈ 𝑉)
203	28	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 0 ∈ V)
204		suppvalfn 8191	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
205	201, 202, 203, 204	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
206		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) = (♯‘(𝑙 supp 0 )))
207		peano2nn 12275	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
208	207	ad3antlr 731	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) ∈ ℕ)
209	208	nnne0d 12313	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑗 + 1) ≠ 0)
210	206, 209	eqnetrrd 3006	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (♯‘(𝑙 supp 0 )) ≠ 0)
211		ovex 7463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 supp 0 ) ∈ V
212		hasheq0 14398	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) = 0 ↔ (𝑙 supp 0 ) = ∅))
213	212	necon3bid 2982	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
214	211, 213	mp1i 13	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ((♯‘(𝑙 supp 0 )) ≠ 0 ↔ (𝑙 supp 0 ) ≠ ∅))
215	210, 214	mpbid 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (𝑙 supp 0 ) ≠ ∅)
216	205, 215	eqnetrrd 3006	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 } ≠ ∅)
217		rabn0 4394	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 } ≠ ∅ ↔ ∃𝑧 ∈ 𝐼 (𝑙‘𝑧) ≠ 0 )
218	216, 217	sylib 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑧 ∈ 𝐼 (𝑙‘𝑧) ≠ 0 )
219	200, 218	reximddv 3168	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑧 ∈ 𝐼 ∃𝑚 ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
220		rexcom 3287	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑧 ∈ 𝐼 ∃𝑚 ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) ↔ ∃𝑚 ∈ (𝐵 ↑_m 𝐼)∃𝑧 ∈ 𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
221	219, 220	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → ∃𝑚 ∈ (𝐵 ↑_m 𝐼)∃𝑧 ∈ 𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))))
222		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))
223		fvoveq1 7453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑚 → (♯‘(ℎ supp 0 )) = (♯‘(𝑚 supp 0 )))
224	223	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (𝑗 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(𝑚 supp 0 ))))
225		eleq1w 2821	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (ℎ ∈ 𝐻 ↔ 𝑚 ∈ 𝐻))
226	224, 225	imbi12d 344	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (ℎ = 𝑚 → ((𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻)))
227	226	rspccva 3620	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
228	227	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) → (𝑗 = (♯‘(𝑚 supp 0 )) → 𝑚 ∈ 𝐻))
229	228	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚 ∈ 𝐻)
230	229	adantllr 719	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ 𝑚 ∈ (𝐵 ↑_m 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚 ∈ 𝐻)
231	230	adantlrr 721	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ 𝑗 = (♯‘(𝑚 supp 0 ))) → 𝑚 ∈ 𝐻)
232	231	adantrr 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑚 ∈ 𝐻)
233		simplrr 778	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑧 ∈ 𝐼)
234	97	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑙 ∈ (𝐵 ↑_m 𝐼))
235	96, 234, 233	mapfvd 8917	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → (𝑙‘𝑧) ∈ 𝐵)
236	68	ad5antr 734	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻)
237		equequ2 2022	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑧 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑧))
238	237	ifbid 4553	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑧 → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = 𝑧, 𝑏, 0 ))
239	238	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )))
240	239	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻))
241		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑧 → (𝑙‘𝑥) = (𝑙‘𝑧))
242	241	eqeq2d 2745	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 = 𝑧 → (𝑏 = (𝑙‘𝑥) ↔ 𝑏 = (𝑙‘𝑧)))
243	242	biimparc 479	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑏 = (𝑙‘𝑧) ∧ 𝑥 = 𝑧) → 𝑏 = (𝑙‘𝑥))
244	243	ifeq1da 4561	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = (𝑙‘𝑧) → if(𝑥 = 𝑧, 𝑏, 0 ) = if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
245	244	mpteq2dv 5249	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = (𝑙‘𝑧) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))
246	245	eleq1d 2823	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = (𝑙‘𝑧) → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻))
247	240, 246	rspc2va 3633	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐼 ∀𝑏 ∈ 𝐵 (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻)
248	233, 235, 236, 247	syl21anc 838	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻)
249		fsuppind.2	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻)) → (𝑥 ∘_f + 𝑦) ∈ 𝐻)
250	249	ralrimivva 3199	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻)
251	250	ad5antr 734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻)
252		ovrspc2v 7456	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 ∈ 𝐻 ∧ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) ∈ 𝐻) ∧ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 ∘_f + 𝑦) ∈ 𝐻) → (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) ∈ 𝐻)
253	232, 248, 251, 252	syl21anc 838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))) ∈ 𝐻)
254	222, 253	eqeltrd 2838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) ∧ (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))))) → 𝑙 ∈ 𝐻)
255	254	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑚 ∈ (𝐵 ↑_m 𝐼) ∧ 𝑧 ∈ 𝐼)) → ((𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) → 𝑙 ∈ 𝐻))
256	255	rexlimdvva 3210	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → (∃𝑚 ∈ (𝐵 ↑_m 𝐼)∃𝑧 ∈ 𝐼 (𝑗 = (♯‘(𝑚 supp 0 )) ∧ 𝑙 = (𝑚 ∘_f + (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))) → 𝑙 ∈ 𝐻))
257	221, 256	mpd 15	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) → 𝑙 ∈ 𝐻)
258	257	exp32 420	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) → (𝑙 ∈ (𝐵 ↑_m 𝐼) → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻)))
259	258	ralrimiv 3142	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) → ∀𝑙 ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻))
260		fvoveq1 7453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = ℎ → (♯‘(𝑙 supp 0 )) = (♯‘(ℎ supp 0 )))
261	260	eqeq2d 2745	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
262		eleq1w 2821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → (𝑙 ∈ 𝐻 ↔ ℎ ∈ 𝐻))
263	261, 262	imbi12d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = ℎ → (((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻) ↔ ((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)))
264	263	cbvralvw 3234	. . . . . . . . . . . . . . 15 ⊢ (∀𝑙 ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(𝑙 supp 0 )) → 𝑙 ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
265	259, 264	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑗 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻)) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((𝑗 + 1) = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
266	9, 12, 15, 18, 86, 265	nnindd 12283	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
267	266	ralrimiva 3143	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
268		ralcom 3286	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
269	267, 268	sylib 218	. . . . . . . . . . 11 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
270		biidd 262	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (♯‘(ℎ supp 0 )) → (ℎ ∈ 𝐻 ↔ ℎ ∈ 𝐻))
271	270	ceqsralv 3519	. . . . . . . . . . . . 13 ⊢ ((♯‘(ℎ supp 0 )) ∈ ℕ → (∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) ↔ ℎ ∈ 𝐻))
272	271	biimpcd 249	. . . . . . . . . . . 12 ⊢ (∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) → ((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
273	272	ralimi 3080	. . . . . . . . . . 11 ⊢ (∀ℎ ∈ (𝐵 ↑_m 𝐼)∀𝑛 ∈ ℕ (𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
274	269, 273	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻))
275		fvoveq1 7453	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑋 → (♯‘(ℎ supp 0 )) = (♯‘(𝑋 supp 0 )))
276	275	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → ((♯‘(ℎ supp 0 )) ∈ ℕ ↔ (♯‘(𝑋 supp 0 )) ∈ ℕ))
277		eleq1 2826	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → (ℎ ∈ 𝐻 ↔ 𝑋 ∈ 𝐻))
278	276, 277	imbi12d 344	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻) ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
279	278	rspcv 3617	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑_m 𝐼) → (∀ℎ ∈ (𝐵 ↑_m 𝐼)((♯‘(ℎ supp 0 )) ∈ ℕ → ℎ ∈ 𝐻) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
280	274, 279	syl5com 31	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ (𝐵 ↑_m 𝐼) → ((♯‘(𝑋 supp 0 )) ∈ ℕ → 𝑋 ∈ 𝐻)))
281	280	com23 86	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(𝑋 supp 0 )) ∈ ℕ → (𝑋 ∈ (𝐵 ↑_m 𝐼) → 𝑋 ∈ 𝐻)))
282	281	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋 ∈ (𝐵 ↑_m 𝐼) → 𝑋 ∈ 𝐻))
283	6, 282	sylbird 260	. . . . . 6 ⊢ ((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → (𝑋:𝐼⟶𝐵 → 𝑋 ∈ 𝐻))
284	283	imp 406	. . . . 5 ⊢ (((𝜑 ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) ∧ 𝑋:𝐼⟶𝐵) → 𝑋 ∈ 𝐻)
285	284	an32s 652	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
286	285	adantlr 715	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) ∈ ℕ) → 𝑋 ∈ 𝐻)
287		ovex 7463	. . . . 5 ⊢ (𝑋 supp 0 ) ∈ V
288		hasheq0 14398	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ V → ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅))
289	287, 288	ax-mp 5	. . . 4 ⊢ ((♯‘(𝑋 supp 0 )) = 0 ↔ (𝑋 supp 0 ) = ∅)
290		ffn 6736	. . . . . . . 8 ⊢ (𝑋:𝐼⟶𝐵 → 𝑋 Fn 𝐼)
291	290	ad2antlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 Fn 𝐼)
292	4	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝐼 ∈ 𝑉)
293	28	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 0 ∈ V)
294		fnsuppeq0 8215	. . . . . . 7 ⊢ ((𝑋 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
295	291, 292, 293, 294	syl3anc 1370	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
296	295	biimpa 476	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 = (𝐼 × { 0 }))
297		fsuppind.0	. . . . . 6 ⊢ (𝜑 → (𝐼 × { 0 }) ∈ 𝐻)
298	297	ad3antrrr 730	. . . . 5 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → (𝐼 × { 0 }) ∈ 𝐻)
299	296, 298	eqeltrd 2838	. . . 4 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (𝑋 supp 0 ) = ∅) → 𝑋 ∈ 𝐻)
300	289, 299	sylan2b 594	. . 3 ⊢ ((((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) ∧ (♯‘(𝑋 supp 0 )) = 0) → 𝑋 ∈ 𝐻)
301		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 finSupp 0 )
302	301	fsuppimpd 9406	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (𝑋 supp 0 ) ∈ Fin)
303		hashcl 14391	. . . . 5 ⊢ ((𝑋 supp 0 ) ∈ Fin → (♯‘(𝑋 supp 0 )) ∈ ℕ₀)
304	302, 303	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (♯‘(𝑋 supp 0 )) ∈ ℕ₀)
305		elnn0 12525	. . . 4 ⊢ ((♯‘(𝑋 supp 0 )) ∈ ℕ₀ ↔ ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
306	304, 305	sylib 218	. . 3 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → ((♯‘(𝑋 supp 0 )) ∈ ℕ ∨ (♯‘(𝑋 supp 0 )) = 0))
307	286, 300, 306	mpjaodan 960	. 2 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → 𝑋 ∈ 𝐻)
308	307	anasss 466	1 ⊢ ((𝜑 ∧ (𝑋:𝐼⟶𝐵 ∧ 𝑋 finSupp 0 )) → 𝑋 ∈ 𝐻)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∃!weu 2565 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∃!wreu 3375 {crab 3432 Vcvv 3477 ∖ cdif 3959 ∅c0 4338 ifcif 4530 {csn 4630 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 ℩crio 7386 (class class class)co 7430 ∘_f cof 7694 supp csupp 8183 ↑_m cmap 8864 Fincfn 8983 finSupp cfsupp 9398 0cc0 11152 1c1 11153 + caddc 11155 ℕcn 12263 ℕ₀cn0 12523 ♯chash 14365 Basecbs 17244 +_gcplusg 17297 0_gc0g 17485 Grpcgrp 18963

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 df-0g 17487 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-grp 18966

This theorem is referenced by: fsuppssind 42579

W3C validator