fsuppind - Mathbox for Steven Nguyen

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Theorem fsuppind 42578

Description: Induction on functions 𝐹:𝐴⟶𝐵 with finite support, or in other words the base set of the free module (see frlmelbas 21665 and frlmplusgval 21673). This theorem is structurally general for polynomial proof usage (see mplelbas 21900 and mpladd 21918). 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 6872	. . . . . . . . . 10 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ V)
4		fsuppind.v	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
5	3, 4	elmapd 8813	. . . . . . . 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 2733	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . 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 3156	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (1 = (♯‘(ℎ supp 0 )) ↔ (♯‘(ℎ supp 0 )) = 1)
20		ovex 7420	. . . . . . . . . . . . . . . . . 18 ⊢ (ℎ supp 0 ) ∈ V
21		euhash1 14385	. . . . . . . . . . . . . . . . . 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 8838	. . . . . . . . . . . . . . . . . . . . 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 6872	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 0 ∈ V
29	28	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → 0 ∈ V)
30		elsuppfn 8149	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((ℎ Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
31	25, 26, 29, 30	syl3anc 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (𝑐 ∈ (ℎ supp 0 ) ↔ (𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
32	31	eubidv 2579	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) → (∃!𝑐 𝑐 ∈ (ℎ supp 0 ) ↔ ∃!𝑐(𝑐 ∈ 𝐼 ∧ (ℎ‘𝑐) ≠ 0 )))
33		df-reu 3355	. . . . . . . . . . . . . . . . . 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 6871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ‘𝑥) ∈ V
37	36, 28	ifex 4539	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), (ℎ‘𝑥), 0 ) ∈ V
38		eqid 2729	. . . . . . . . . . . . . . . . . . . . . 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 6661	. . . . . . . . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ↔ 𝑣 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
42		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑣 → (ℎ‘𝑥) = (ℎ‘𝑣))
43	41, 42	ifbieq1d 4513	. . . . . . . . . . . . . . . . . . . . . . 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 6973	. . . . . . . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ℎ ∈ (𝐵 ↑_m 𝐼)) ∧ ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) ∧ 𝑣 ∈ 𝐼) → ∃!𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )
49		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑐 = 𝑣 → (ℎ‘𝑐) = (ℎ‘𝑣))
50	49	neeq1d 2984	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑐 = 𝑣 → ((ℎ‘𝑐) ≠ 0 ↔ (ℎ‘𝑣) ≠ 0 ))
51	50	riota2 7369	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2978	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ( 0 ≠ (ℎ‘𝑣) ↔ (ℎ‘𝑣) ≠ 0 )
54		eqcom 2736	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2943	. . . . . . . . . . . . . . . . . . . . . . 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 4525	. . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . 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 7006	. . . . . . . . . . . . . . . . . . 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 7361	. . . . . . . . . . . . . . . . . . . . 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 8822	. . . . . . . . . . . . . . . . . . . . . 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 7057	. . . . . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (𝑥 = 𝑎 ↔ 𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
71	70	ifbid 4512	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ), 𝑏, 0 ))
72	71	mpteq2dv 5201	. . . . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = (℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 ) → (ℎ‘𝑥) = (ℎ‘(℩𝑐 ∈ 𝐼 (ℎ‘𝑐) ≠ 0 )))
75	74	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . 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 4520	. . . . . . . . . . . . . . . . . . . . . . 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 5201	. . . . . . . . . . . . . . . . . . . . . 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 2813	. . . . . . . . . . . . . . . . . . . . 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 3600	. . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . 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 3125	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(1 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
87		fvoveq1 7410	. . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . 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 7394	. . . . . . . . . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . . . . . . . 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 18897	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐵 ↑_m 𝐼) = (𝐵 ↑_m 𝐼)
97		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 8852	. . . . . . . . . . . . . . . . . . . . . . . . 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 4535	. . . . . . . . . . . . . . . . . . . . . . . 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 7087	. . . . . . . . . . . . . . . . . . . . . . 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 8813	. . . . . . . . . . . . . . . . . . . . . . 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 7422	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙 supp 0 ) ∈ V)
109		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑧 ∈ 𝐼)
110		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → (𝑙‘𝑧) ≠ 0 )
111		elmapfn 8838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8149	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑧 ∈ (𝑙 supp 0 ) ↔ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )))
117	113, 114, 115, 116	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ)
120	119	nnnn0d 12503	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ (𝑙 ∈ (𝐵 ↑_m 𝐼) ∧ (𝑗 + 1) = (♯‘(𝑙 supp 0 )))) ∧ (𝑧 ∈ 𝐼 ∧ (𝑙‘𝑧) ≠ 0 )) → 𝑗 ∈ ℕ₀)
121		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . . . . . . 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 14471	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1375	. . . . . . . . . . . . . . . . . . . . . . . 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 4750	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑣 ∈ ((𝑙 supp 0 ) ∖ {𝑧}) ↔ (𝑣 ∈ (𝑙 supp 0 ) ∧ 𝑣 ≠ 𝑧))
127		fvex 6871	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑙‘𝑥) ∈ V
128	28, 127	ifex 4539	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) ∈ V
129		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥))) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)))
130	128, 129	fnmpti 6661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4497	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (¬ 𝑣 = 𝑧 → if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) = (𝑙‘𝑣))
140	139	eqeq1d 2731	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3018	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2025	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑥 = 𝑧 ↔ 𝑣 = 𝑧))
154		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑥 = 𝑣 → (𝑙‘𝑥) = (𝑙‘𝑣))
155	153, 154	ifbieq2d 4515	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, 0 , (𝑙‘𝑥)) = if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)))
156	155, 129, 128	fvmpt3i 6973	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2984	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8149	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑣 ∈ (𝑙 supp 0 ) ↔ (𝑣 ∈ 𝐼 ∧ (𝑙‘𝑣) ≠ 0 )))
162	160, 132, 133, 161	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2727	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . . . 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 4539	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) ∈ V
172		eqid 2729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
173	171, 172	fnmpti 6661	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4190	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝐼 ∩ 𝐼) = 𝐼
176	131, 174, 132, 132, 175	offn 7666	. . . . . . . . . . . . . . . . . . . . . . . . 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 4513	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑣 → if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ) = if(𝑣 = 𝑧, (𝑙‘𝑣), 0 ))
178	177, 172, 171	fvmpt3i 6973	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7664	. . . . . . . . . . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 8852	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 18899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ( 0 + (𝑙‘𝑣)) = (𝑙‘𝑣))
188	1, 186, 27	grprid 18900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝐺 ∈ Grp ∧ (𝑙‘𝑣) ∈ 𝐵) → ((𝑙‘𝑣) + 0 ) = (𝑙‘𝑣))
189	187, 188	ifeq12d 4510	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7489	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (if(𝑣 = 𝑧, 0 , (𝑙‘𝑣)) + if(𝑣 = 𝑧, (𝑙‘𝑣), 0 )) = if(𝑣 = 𝑧, ( 0 + (𝑙‘𝑣)), ((𝑙‘𝑣) + 0 ))
192		ifid 4529	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣)) = (𝑙‘𝑣)
193	192	eqcomi 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑙‘𝑣) = if(𝑣 = 𝑧, (𝑙‘𝑣), (𝑙‘𝑣))
194	190, 191, 193	3eqtr4g 2789	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . . . . . 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 7006	. . . . . . . . . . . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . . . . . . . . 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 8147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑙 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → (𝑙 supp 0 ) = {𝑧 ∈ 𝐼 ∣ (𝑙‘𝑧) ≠ 0 })
205	201, 202, 203, 204	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . . . . 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 12198	. . . . . . . . . . . . . . . . . . . . . . . . . 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 12236	. . . . . . . . . . . . . . . . . . . . . . . 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 2993	. . . . . . . . . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑙 supp 0 ) ∈ V
212		hasheq0 14328	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑙 supp 0 ) ∈ V → ((♯‘(𝑙 supp 0 )) = 0 ↔ (𝑙 supp 0 ) = ∅))
213	212	necon3bid 2969	. . . . . . . . . . . . . . . . . . . . . . . 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 2993	. . . . . . . . . . . . . . . . . . . . 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 4352	. . . . . . . . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . . . . . . . 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 3266	. . . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . . 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 7410	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (ℎ = 𝑚 → (♯‘(ℎ supp 0 )) = (♯‘(𝑚 supp 0 )))
224	223	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℎ = 𝑚 → (𝑗 = (♯‘(ℎ supp 0 )) ↔ 𝑗 = (♯‘(𝑚 supp 0 ))))
225		eleq1w 2811	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3587	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . . 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 8852	. . . . . . . . . . . . . . . . . . . . . . 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 2026	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 = 𝑧 → (𝑥 = 𝑎 ↔ 𝑥 = 𝑧))
238	237	ifbid 4512	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 = 𝑧 → if(𝑥 = 𝑎, 𝑏, 0 ) = if(𝑥 = 𝑧, 𝑏, 0 ))
239	238	mpteq2dv 5201	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 = 𝑧 → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )))
240	239	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑎 = 𝑧 → ((𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑎, 𝑏, 0 )) ∈ 𝐻 ↔ (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) ∈ 𝐻))
241		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑥 = 𝑧 → (𝑙‘𝑥) = (𝑙‘𝑧))
242	241	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4520	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = (𝑙‘𝑧) → if(𝑥 = 𝑧, 𝑏, 0 ) = if(𝑥 = 𝑧, (𝑙‘𝑥), 0 ))
245	244	mpteq2dv 5201	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = (𝑙‘𝑧) → (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, 𝑏, 0 )) = (𝑥 ∈ 𝐼 ↦ if(𝑥 = 𝑧, (𝑙‘𝑥), 0 )))
246	245	eleq1d 2813	. . . . . . . . . . . . . . . . . . . . . . . 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 3600	. . . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . . 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 3180	. . . . . . . . . . . . . . . . . . . . . . 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 7413	. . . . . . . . . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . . . . . 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 3194	. . . . . . . . . . . . . . . . . 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 3124	. . . . . . . . . . . . . . 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 7410	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = ℎ → (♯‘(𝑙 supp 0 )) = (♯‘(ℎ supp 0 )))
261	260	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = ℎ → ((𝑗 + 1) = (♯‘(𝑙 supp 0 )) ↔ (𝑗 + 1) = (♯‘(ℎ supp 0 ))))
262		eleq1w 2811	. . . . . . . . . . . . . . . . 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 3215	. . . . . . . . . . . . . . 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 12206	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
267	266	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀ℎ ∈ (𝐵 ↑_m 𝐼)(𝑛 = (♯‘(ℎ supp 0 )) → ℎ ∈ 𝐻))
268		ralcom 3265	. . . . . . . . . . . 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 3488	. . . . . . . . . . . . 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 3066	. . . . . . . . . . 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 7410	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑋 → (♯‘(ℎ supp 0 )) = (♯‘(𝑋 supp 0 )))
276	275	eleq1d 2813	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑋 → ((♯‘(ℎ supp 0 )) ∈ ℕ ↔ (♯‘(𝑋 supp 0 )) ∈ ℕ))
277		eleq1 2816	. . . . . . . . . . . 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 3584	. . . . . . . . . 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 7420	. . . . 5 ⊢ (𝑋 supp 0 ) ∈ V
288		hasheq0 14328	. . . . 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 6688	. . . . . . . 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 8171	. . . . . . 7 ⊢ ((𝑋 Fn 𝐼 ∧ 𝐼 ∈ 𝑉 ∧ 0 ∈ V) → ((𝑋 supp 0 ) = ∅ ↔ 𝑋 = (𝐼 × { 0 })))
295	291, 292, 293, 294	syl3anc 1373	. . . . . 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 2828	. . . 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 9320	. . . . 5 ⊢ (((𝜑 ∧ 𝑋:𝐼⟶𝐵) ∧ 𝑋 finSupp 0 ) → (𝑋 supp 0 ) ∈ Fin)
303		hashcl 14321	. . . . 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 12444	. . . 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 1540 ∈ wcel 2109 ∃!weu 2561 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∃!wreu 3352 {crab 3405 Vcvv 3447 ∖ cdif 3911 ∅c0 4296 ifcif 4488 {csn 4589 class class class wbr 5107 ↦ cmpt 5188 × cxp 5636 Fn wfn 6506 ⟶wf 6507 ‘cfv 6511 ℩crio 7343 (class class class)co 7387 ∘_f cof 7651 supp csupp 8139 ↑_m cmap 8799 Fincfn 8918 finSupp cfsupp 9312 0cc0 11068 1c1 11069 + caddc 11071 ℕcn 12186 ℕ₀cn0 12442 ♯chash 14295 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 Grpcgrp 18865

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-oadd 8438 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-dju 9854 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-hash 14296 df-0g 17404 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-grp 18868

This theorem is referenced by: fsuppssind 42581

W3C validator