Theorem selvply1rhmlemb 34221

Description: Lemma for selvply1rhm 34227. (Contributed by Thierry Arnoux, 4-May-2026.)

Hypotheses

Ref	Expression
selvply1rhmlema.1	⊢ 𝐵 = (Base‘𝑃)
selvply1rhmlema.2	⊢ 𝑃 = ({𝑋} mPoly 𝑅)
selvply1rhmlema.3	⊢ · = (.r‘𝑃)
selvply1rhmlema.4	⊢ × = (.r‘𝑄)
selvply1rhmlema.5	⊢ 𝑄 = (Poly1‘𝑅)
selvply1rhmlema.6	⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})))
selvply1rhmlema.7	⊢ (𝜑 → 𝑋 ∈ 𝑉)
selvply1rhmlema.8	⊢ (𝜑 → 𝑅 ∈ Ring)
selvply1rhmlema.9	⊢ (𝜑 → 𝐹 ∈ 𝐵)
selvply1rhmlemb.10	⊢ (𝜑 → 𝐺 ∈ 𝐵)

Assertion

Ref	Expression
selvply1rhmlemb	⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺)))

Distinct variable groups:   · ,𝑓   𝐵,𝑓   𝑓,𝐹,𝑛   𝑅,𝑛   𝑓,𝑋,𝑛   𝜑,𝑓,𝑛   × ,𝑓   · ,𝑛   𝑓,𝐺,𝑛   𝑓,𝑀,𝑛

Allowed substitution hints:   𝐵(𝑛)   𝑃(𝑓, 𝑛)   𝑄(𝑓, 𝑛)   𝑅(𝑓)   × (𝑛)   𝑉(𝑓, 𝑛)

Proof of Theorem selvply1rhmlemb

Dummy variables 𝑚 ℎ 𝑖 𝑗 𝑥 𝑔 𝑙 𝑘 𝑜 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		selvply1rhmlema.6	. 2 ⊢ 𝑀 = (𝑓 ∈ 𝐵 ↦ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})))
2		fveq1 6907	. . . 4 ⊢ (𝑓 = (𝐹 · 𝐺) → (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}) = ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩}))
3	2	mpteq2dv 5202	. . 3 ⊢ (𝑓 = (𝐹 · 𝐺) → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩})))
4		selvply1rhmlema.2	. . . . . . . 8 ⊢ 𝑃 = ({𝑋} mPoly 𝑅)
5		selvply1rhmlema.1	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑃)
6		eqid 2763	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
7		selvply1rhmlema.3	. . . . . . . 8 ⊢ · = (.r‘𝑃)
8		eqid 2763	. . . . . . . . 9 ⊢ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} = {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}
9	8	psrbasfsupp 34213	. . . . . . . 8 ⊢ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} = {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ (◡𝑔 “ ℕ) ∈ Fin}
10		selvply1rhmlema.9	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
11		selvply1rhmlemb.10	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
12	4, 5, 6, 7, 9, 10, 11	mplmul 22348	. . . . . . 7 ⊢ (𝜑 → (𝐹 · 𝐺) = (𝑚 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ↦ (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗)))))))
13	12	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝐹 · 𝐺) = (𝑚 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ↦ (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗)))))))
14		breq2 5110	. . . . . . . . . 10 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑙 ∘r ≤ 𝑚 ↔ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
15	14	rabbidv 3423	. . . . . . . . 9 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} = {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}})
16		fvoveq1 7466	. . . . . . . . . 10 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝐺‘(𝑚 ∘f − 𝑗)) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)))
17	16	oveq2d 7459	. . . . . . . . 9 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗))) = ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))
18	15, 17	mpteq12dv 5195	. . . . . . . 8 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗)))) = (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)))))
19	18	oveq2d 7459	. . . . . . 7 ⊢ (𝑚 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗))))) = (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))))
20		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑗((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
21		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
22		eqid 2763	. . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅)
23		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → (𝐹‘𝑗) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
24		oveq2 7451	. . . . . . . . . . 11 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) = ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))
25	24	fveq2d 6912	. . . . . . . . . 10 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
26	23, 25	oveq12d 7461	. . . . . . . . 9 ⊢ (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} → ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))) = ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))
27		selvply1rhmlema.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Ring)
28	27	ringcmnd 20543	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CMnd)
29	28	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑅 ∈ CMnd)
30		eqid 2763	. . . . . . . . . . 11 ⊢ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} = {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}
31		ovexd 7478	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℕ0 ↑m {𝑋}) ∈ V)
32	8, 31	rabexd 5305	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∈ V)
33	30, 32	rabexd 5305	. . . . . . . . . 10 ⊢ (𝜑 → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ∈ V)
34	33	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ∈ V)
35		fvexd 6923	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (0g‘𝑅) ∈ V)
36	32	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∈ V)
37		ssrab2 4008	. . . . . . . . . . 11 ⊢ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}
38	37	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
39	4, 21, 5, 9, 11	mplelf 22335	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
40	39	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝐺:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
41		breq1 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = {⟨𝑋, (𝑛‘∅)⟩} → (𝑔 finSupp 0 ↔ {⟨𝑋, (𝑛‘∅)⟩} finSupp 0))
42		nn0ex 12639	. . . . . . . . . . . . . . . . 17 ⊢ ℕ0 ∈ V
43	42	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ℕ0 ∈ V)
44		snex 5401	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ∈ V
45	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑋} ∈ V)
46		selvply1rhmlema.7	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
47	46	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑋 ∈ 𝑉)
48		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛 ∈ (ℕ0 ↑m 1o))
49	48	elmaprd 8898	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛:1o⟶ℕ0)
50		0lt1o 8540	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ 1o
51	50	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ∅ ∈ 1o)
52	49, 51	ffvelcdmd 7109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛‘∅) ∈ ℕ0)
53	47, 52	fsnd 6892	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩}:{𝑋}⟶ℕ0)
54	43, 45, 53	elmapdd 8889	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩} ∈ (ℕ0 ↑m {𝑋}))
55		snfi 9099	. . . . . . . . . . . . . . . . 17 ⊢ {𝑋} ∈ Fin
56	55	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑋} ∈ Fin)
57		c0ex 11327	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
58	57	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 0 ∈ V)
59	53, 56, 58	fdmfifsupp 9394	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩} finSupp 0)
60	41, 54, 59	elrabd 3650	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨𝑋, (𝑛‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
61	60	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨𝑋, (𝑛‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
62		ssrab2 4008	. . . . . . . . . . . . . . . . 17 ⊢ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ⊆ (ℕ0 ↑m {𝑋})
63	37, 62	sstri 3922	. . . . . . . . . . . . . . . 16 ⊢ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ (ℕ0 ↑m {𝑋})
64	63	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ⊆ (ℕ0 ↑m {𝑋}))
65	64	sselda 3913	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∈ (ℕ0 ↑m {𝑋}))
66	65	elmaprd 8898	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗:{𝑋}⟶ℕ0)
67		breq1 5109	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑗 → (𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ↔ 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
68		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}})
69	67, 68	elrabrd 3651	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩})
70	9	psrbagcon 22263	. . . . . . . . . . . . 13 ⊢ (({⟨𝑋, (𝑛‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∧ 𝑗:{𝑋}⟶ℕ0 ∧ 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∧ ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
71	61, 66, 69, 70	syl3anc 1399	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∧ ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
72	71	simpld 500	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗) ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
73	40, 72	ffvelcdmd 7109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)) ∈ (Base‘𝑅))
74	4, 21, 5, 9, 10	mplelf 22335	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
75	74	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝐹:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
76	4, 5, 22, 10	mplelsfi 22332	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
77	76	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝐹 finSupp (0g‘𝑅))
78	27	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
79		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑥 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
80	21, 6, 22, 78, 79	ringlzd 20556	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑥 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑥) = (0g‘𝑅))
81	35, 35, 36, 38, 73, 75, 77, 80	fisuppov1 33347	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗)))) finSupp (0g‘𝑅))
82		ssidd 3936	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (Base‘𝑅) ⊆ (Base‘𝑅))
83	27	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑅 ∈ Ring)
84	74	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝐹:{𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0}⟶(Base‘𝑅))
85	38	sselda 3913	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑗 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
86	84, 85	ffvelcdmd 7109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝐹‘𝑗) ∈ (Base‘𝑅))
87	21, 6, 83, 86, 73	ringcld 20514	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))) ∈ (Base‘𝑅))
88		breq1 5109	. . . . . . . . . 10 ⊢ (𝑙 = {⟨𝑋, (𝑖‘∅)⟩} → (𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ↔ {⟨𝑋, (𝑖‘∅)⟩} ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}))
89		breq1 5109	. . . . . . . . . . 11 ⊢ (𝑔 = {⟨𝑋, (𝑖‘∅)⟩} → (𝑔 finSupp 0 ↔ {⟨𝑋, (𝑖‘∅)⟩} finSupp 0))
90	42	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ℕ0 ∈ V)
91	44	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {𝑋} ∈ V)
92	47	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑋 ∈ 𝑉)
93		ssrab2 4008	. . . . . . . . . . . . . . . . 17 ⊢ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ⊆ (ℕ0 ↑m 1o)
94	93	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ⊆ (ℕ0 ↑m 1o))
95	94	sselda 3913	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∈ (ℕ0 ↑m 1o))
96	95	elmaprd 8898	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖:1o⟶ℕ0)
97	50	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∅ ∈ 1o)
98	96, 97	ffvelcdmd 7109	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ∈ ℕ0)
99	92, 98	fsnd 6892	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩}:{𝑋}⟶ℕ0)
100	90, 91, 99	elmapdd 8889	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∈ (ℕ0 ↑m {𝑋}))
101	55	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {𝑋} ∈ Fin)
102	57	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 0 ∈ V)
103	99, 101, 102	fdmfifsupp 9394	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} finSupp 0)
104	89, 100, 103	elrabd 3650	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0})
105		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑛 ∈ (ℕ0 ↑m 1o))
106		breq1 5109	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑖 → (𝑘 ∘r ≤ 𝑛 ↔ 𝑖 ∘r ≤ 𝑛))
107		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛})
108	106, 107	elrabrd 3651	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∘r ≤ 𝑛)
109		elmapfn 8915	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (ℕ0 ↑m 1o) → 𝑖 Fn 1o)
110	109	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 𝑖 Fn 1o)
111		elmapfn 8915	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → 𝑛 Fn 1o)
112	111	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 𝑛 Fn 1o)
113		1oex 8514	. . . . . . . . . . . . . . 15 ⊢ 1o ∈ V
114	113	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 1o ∈ V)
115		inidm 4126	. . . . . . . . . . . . . 14 ⊢ (1o ∩ 1o) = 1o
116		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) ∧ ∅ ∈ 1o) → (𝑖‘∅) = (𝑖‘∅))
117		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) ∧ ∅ ∈ 1o) → (𝑛‘∅) = (𝑛‘∅))
118	110, 112, 114, 114, 115, 116, 117	ofrval 7729	. . . . . . . . . . . . 13 ⊢ (((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∘r ≤ 𝑛 ∧ ∅ ∈ 1o) → (𝑖‘∅) ≤ (𝑛‘∅))
119	105, 95, 108, 97, 118	syl211anc 1404	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ≤ (𝑛‘∅))
120	119	ralrimivw 3162	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∀𝑥 ∈ {𝑋} (𝑖‘∅) ≤ (𝑛‘∅))
121	99	ffnd 6732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} Fn {𝑋})
122	53	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑛‘∅)⟩}:{𝑋}⟶ℕ0)
123	122	ffnd 6732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
124		inidm 4126	. . . . . . . . . . . 12 ⊢ ({𝑋} ∩ {𝑋}) = {𝑋}
125		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → 𝑥 ∈ {𝑋})
126	125	elsnd 4602	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → 𝑥 = 𝑋)
127	126	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑥) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋))
128		fvsng 7208	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑖‘∅) ∈ ℕ0) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
129	92, 98, 128	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
130	129	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
131	127, 130	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑥) = (𝑖‘∅))
132	126	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑥) = ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋))
133	52	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛‘∅) ∈ ℕ0)
134		fvsng 7208	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑛‘∅) ∈ ℕ0) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
135	92, 133, 134	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
136	135	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
137	132, 136	eqtrd 2798	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑥 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑥) = (𝑛‘∅))
138	121, 123, 91, 91, 124, 131, 137	ofrfval 7727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑖‘∅)⟩} ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ↔ ∀𝑥 ∈ {𝑋} (𝑖‘∅) ≤ (𝑛‘∅)))
139	120, 138	mpbird 260	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩})
140	88, 104, 139	elrabd 3650	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}})
141		breq1 5109	. . . . . . . . . . 11 ⊢ (𝑘 = {⟨∅, (𝑗‘𝑋)⟩} → (𝑘 ∘r ≤ 𝑛 ↔ {⟨∅, (𝑗‘𝑋)⟩} ∘r ≤ 𝑛))
142	42	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ℕ0 ∈ V)
143	113	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 1o ∈ V)
144		df1o2 8511	. . . . . . . . . . . . . . 15 ⊢ 1o = {∅}
145	144	eqcomi 2772	. . . . . . . . . . . . . 14 ⊢ {∅} = 1o
146	145	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {∅} = 1o)
147		0ex 5264	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
148	147	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∅ ∈ V)
149		snidg 4621	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
150	46, 149	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ {𝑋})
151	150	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑋 ∈ {𝑋})
152	66, 151	ffvelcdmd 7109	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝑗‘𝑋) ∈ ℕ0)
153	148, 152	fsnd 6892	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩}:{∅}⟶ℕ0)
154	146, 153	feq2dd 6717	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩}:1o⟶ℕ0)
155	142, 143, 154	elmapdd 8889	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} ∈ (ℕ0 ↑m 1o))
156		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑛 ∈ (ℕ0 ↑m 1o))
157	47	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑋 ∈ 𝑉)
158	156, 157	jca 521	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉))
159		elmapfn 8915	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℕ0 ↑m {𝑋}) → 𝑗 Fn {𝑋})
160	159	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) → 𝑗 Fn {𝑋})
161		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ 𝑉)
162		elmapi 8897	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → 𝑛:1o⟶ℕ0)
163	50	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → ∅ ∈ 1o)
164	162, 163	ffvelcdmd 7109	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) → (𝑛‘∅) ∈ ℕ0)
165	164	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → (𝑛‘∅) ∈ ℕ0)
166	161, 165	fsnd 6892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → {⟨𝑋, (𝑛‘∅)⟩}:{𝑋}⟶ℕ0)
167	166	ffnd 6732	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
168	167	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
169	44	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) → {𝑋} ∈ V)
170		eqidd 2764	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) ∧ 𝑋 ∈ {𝑋}) → (𝑗‘𝑋) = (𝑗‘𝑋))
171	161, 165, 134	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
172	171	ad2antlr 740	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) ∧ 𝑋 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
173	160, 168, 169, 169, 124, 170, 172	ofrval 7729	. . . . . . . . . . . . . . 15 ⊢ (((𝑗 ∈ (ℕ0 ↑m {𝑋}) ∧ (𝑛 ∈ (ℕ0 ↑m 1o) ∧ 𝑋 ∈ 𝑉)) ∧ 𝑗 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩} ∧ 𝑋 ∈ {𝑋}) → (𝑗‘𝑋) ≤ (𝑛‘∅))
174	65, 158, 69, 151, 173	syl211anc 1404	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → (𝑗‘𝑋) ≤ (𝑛‘∅))
175		fveq2 6908	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 = ∅ → (𝑛‘𝑜) = (𝑛‘∅))
176	175	breq2d 5118	. . . . . . . . . . . . . . 15 ⊢ (𝑜 = ∅ → ((𝑗‘𝑋) ≤ (𝑛‘𝑜) ↔ (𝑗‘𝑋) ≤ (𝑛‘∅)))
177	147, 176	ralsn 4642	. . . . . . . . . . . . . 14 ⊢ (∀𝑜 ∈ {∅} (𝑗‘𝑋) ≤ (𝑛‘𝑜) ↔ (𝑗‘𝑋) ≤ (𝑛‘∅))
178	174, 177	sylibr 237	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∀𝑜 ∈ {∅} (𝑗‘𝑋) ≤ (𝑛‘𝑜))
179	144	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 1o = {∅})
180	178, 179	raleqtrrdv 3327	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∀𝑜 ∈ 1o (𝑗‘𝑋) ≤ (𝑛‘𝑜))
181	154	ffnd 6732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} Fn 1o)
182	111	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → 𝑛 Fn 1o)
183		elsni 4601	. . . . . . . . . . . . . . . . 17 ⊢ (𝑜 ∈ {∅} → 𝑜 = ∅)
184	183, 144	eleq2s 2882	. . . . . . . . . . . . . . . 16 ⊢ (𝑜 ∈ 1o → 𝑜 = ∅)
185	184	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → 𝑜 = ∅)
186	185	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → ({⟨∅, (𝑗‘𝑋)⟩}‘𝑜) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅))
187	152	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → (𝑗‘𝑋) ∈ ℕ0)
188		fvsng 7208	. . . . . . . . . . . . . . 15 ⊢ ((∅ ∈ V ∧ (𝑗‘𝑋) ∈ ℕ0) → ({⟨∅, (𝑗‘𝑋)⟩}‘∅) = (𝑗‘𝑋))
189	147, 187, 188	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → ({⟨∅, (𝑗‘𝑋)⟩}‘∅) = (𝑗‘𝑋))
190	186, 189	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → ({⟨∅, (𝑗‘𝑋)⟩}‘𝑜) = (𝑗‘𝑋))
191		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑜 ∈ 1o) → (𝑛‘𝑜) = (𝑛‘𝑜))
192	181, 182, 143, 143, 115, 190, 191	ofrfval 7727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ({⟨∅, (𝑗‘𝑋)⟩} ∘r ≤ 𝑛 ↔ ∀𝑜 ∈ 1o (𝑗‘𝑋) ≤ (𝑛‘𝑜)))
193	180, 192	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} ∘r ≤ 𝑛)
194	141, 155, 193	elrabd 3650	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → {⟨∅, (𝑗‘𝑋)⟩} ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛})
195		eqcom 2770	. . . . . . . . . . . . 13 ⊢ ((𝑗‘𝑋) = (𝑖‘∅) ↔ (𝑖‘∅) = (𝑗‘𝑋))
196	195	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑗‘𝑋) = (𝑖‘∅) ↔ (𝑖‘∅) = (𝑗‘𝑋)))
197	129	adantlr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
198	197	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑗‘𝑋) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) ↔ (𝑗‘𝑋) = (𝑖‘∅)))
199	152	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑗‘𝑋) ∈ ℕ0)
200	147, 199, 188	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ({⟨∅, (𝑗‘𝑋)⟩}‘∅) = (𝑗‘𝑋))
201	200	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑖‘∅) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅) ↔ (𝑖‘∅) = (𝑗‘𝑋)))
202	196, 198, 201	3bitr4d 314	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑗‘𝑋) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) ↔ (𝑖‘∅) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅)))
203	157	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑋 ∈ 𝑉)
204		eqid 2763	. . . . . . . . . . . 12 ⊢ {𝑋} = {𝑋}
205	66	adantr 486	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑗:{𝑋}⟶ℕ0)
206	205	ffnd 6732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑗 Fn {𝑋})
207	121	adantlr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨𝑋, (𝑖‘∅)⟩} Fn {𝑋})
208	203, 204, 206, 207	fsneq 7057	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} ↔ (𝑗‘𝑋) = ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋)))
209	147	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∅ ∈ V)
210	96	adantlr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖:1o⟶ℕ0)
211	210	ffnd 6732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 Fn 1o)
212	181	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {⟨∅, (𝑗‘𝑋)⟩} Fn 1o)
213	209, 144, 211, 212	fsneq 7057	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖 = {⟨∅, (𝑗‘𝑋)⟩} ↔ (𝑖‘∅) = ({⟨∅, (𝑗‘𝑋)⟩}‘∅)))
214	202, 208, 213	3bitr4d 314	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑗 = {⟨𝑋, (𝑖‘∅)⟩} ↔ 𝑖 = {⟨∅, (𝑗‘𝑋)⟩}))
215	194, 214	reu6dv 33141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}}) → ∃!𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}𝑗 = {⟨𝑋, (𝑖‘∅)⟩})
216	20, 21, 22, 26, 29, 34, 81, 82, 87, 140, 215	gsummptfsf1o 33691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))))
217	93	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ⊆ (ℕ0 ↑m 1o))
218	217	sselda 3913	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 ∈ (ℕ0 ↑m 1o))
219		fveq1 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (𝑛‘∅) = (𝑖‘∅))
220	219	opeq2d 4850	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → ⟨𝑋, (𝑛‘∅)⟩ = ⟨𝑋, (𝑖‘∅)⟩)
221	220	sneqd 4596	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑖 → {⟨𝑋, (𝑛‘∅)⟩} = {⟨𝑋, (𝑖‘∅)⟩})
222	221	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑖 → (𝐹‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
223		fveq1 6907	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝐹 → (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐹‘{⟨𝑋, (𝑛‘∅)⟩}))
224	223	mpteq2dv 5202	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝐹 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})))
225		ovexd 7478	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℕ0 ↑m 1o) ∈ V)
226	225	mptexd 7253	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})) ∈ V)
227	1, 224, 10, 226	fvmptd3 7040	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀‘𝐹) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})))
228	227	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → (𝑀‘𝐹) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐹‘{⟨𝑋, (𝑛‘∅)⟩})))
229		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → 𝑖 ∈ (ℕ0 ↑m 1o))
230		fvexd 6923	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}) ∈ V)
231	222, 228, 229, 230	fvmptd4 7041	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ (ℕ0 ↑m 1o)) → ((𝑀‘𝐹)‘𝑖) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
232	218, 231	syldan 603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑀‘𝐹)‘𝑖) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
233	232	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑀‘𝐹)‘𝑖) = (𝐹‘{⟨𝑋, (𝑖‘∅)⟩}))
234		fveq1 6907	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝐺 → (𝑓‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐺‘{⟨𝑋, (𝑛‘∅)⟩}))
235	234	mpteq2dv 5202	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐺 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})))
236	225	mptexd 7253	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})) ∈ V)
237	1, 235, 11, 236	fvmptd3 7040	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})))
238		fveq1 6907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑚 → (𝑛‘∅) = (𝑚‘∅))
239	238	opeq2d 4850	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑚 → ⟨𝑋, (𝑛‘∅)⟩ = ⟨𝑋, (𝑚‘∅)⟩)
240	239	sneqd 4596	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑚 → {⟨𝑋, (𝑛‘∅)⟩} = {⟨𝑋, (𝑚‘∅)⟩})
241	240	fveq2d 6912	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑚 → (𝐺‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝐺‘{⟨𝑋, (𝑚‘∅)⟩}))
242	241	cbvmptv 5212	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑚 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑚‘∅)⟩}))
243	237, 242	eqtrdi 2814	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀‘𝐺) = (𝑚 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑚‘∅)⟩})))
244	243	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑀‘𝐺) = (𝑚 ∈ (ℕ0 ↑m 1o) ↦ (𝐺‘{⟨𝑋, (𝑚‘∅)⟩})))
245		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑚 = (𝑛 ∘f − 𝑖))
246	245	fveq1d 6910	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) = ((𝑛 ∘f − 𝑖)‘∅))
247	50	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ∅ ∈ 1o)
248	111	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛 Fn 1o)
249	248	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑛 Fn 1o)
250	95, 109	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 Fn 1o)
251	250	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑖 Fn 1o)
252	113	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 1o ∈ V)
253		eqidd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ ∅ ∈ 1o) → (𝑛‘∅) = (𝑛‘∅))
254		eqidd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ ∅ ∈ 1o) → (𝑖‘∅) = (𝑖‘∅))
255	249, 251, 252, 252, 115, 253, 254	ofval 7728	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ ∅ ∈ 1o) → ((𝑛 ∘f − 𝑖)‘∅) = ((𝑛‘∅) − (𝑖‘∅)))
256	247, 255	mpdan 700	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ((𝑛 ∘f − 𝑖)‘∅) = ((𝑛‘∅) − (𝑖‘∅)))
257	246, 256	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) = ((𝑛‘∅) − (𝑖‘∅)))
258	92	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑋 ∈ 𝑉)
259		fvexd 6923	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) ∈ V)
260		fvsng 7208	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑚‘∅) ∈ V) → ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (𝑚‘∅))
261	258, 259, 260	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (𝑚‘∅))
262	258, 149	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → 𝑋 ∈ {𝑋})
263	123	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑛‘∅)⟩} Fn {𝑋})
264	121	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑖‘∅)⟩} Fn {𝑋})
265	44	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {𝑋} ∈ V)
266	135	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ 𝑋 ∈ {𝑋}) → ({⟨𝑋, (𝑛‘∅)⟩}‘𝑋) = (𝑛‘∅))
267	129	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ 𝑋 ∈ {𝑋}) → ({⟨𝑋, (𝑖‘∅)⟩}‘𝑋) = (𝑖‘∅))
268	263, 264, 265, 265, 124, 266, 267	ofval 7728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) ∧ 𝑋 ∈ {𝑋}) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋) = ((𝑛‘∅) − (𝑖‘∅)))
269	262, 268	mpdan 700	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋) = ((𝑛‘∅) − (𝑖‘∅)))
270	257, 261, 269	3eqtr4d 2808	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋))
271		elsni 4601	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ {(𝑛‘∅)} → 𝑥 = (𝑛‘∅))
272	271	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → 𝑥 = (𝑛‘∅))
273	272	oveq1d 7458	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → (𝑥 − 𝑦) = ((𝑛‘∅) − 𝑦))
274		fznn0sub2 13796	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0...(𝑛‘∅)) → ((𝑛‘∅) − 𝑦) ∈ (0...(𝑛‘∅)))
275	274	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → ((𝑛‘∅) − 𝑦) ∈ (0...(𝑛‘∅)))
276	273, 275	eqeltrd 2864	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅))) → (𝑥 − 𝑦) ∈ (0...(𝑛‘∅)))
277	276	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ (𝑥 ∈ {(𝑛‘∅)} ∧ 𝑦 ∈ (0...(𝑛‘∅)))) → (𝑥 − 𝑦) ∈ (0...(𝑛‘∅)))
278		fvex 6921	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛‘∅) ∈ V
279	147, 278	f1osn 6889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {⟨∅, (𝑛‘∅)⟩}:{∅}⟶1-1onto{(𝑛‘∅)}
280		f1of 6846	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ({⟨∅, (𝑛‘∅)⟩}:{∅}⟶1-1onto{(𝑛‘∅)} → {⟨∅, (𝑛‘∅)⟩}:{∅}⟶{(𝑛‘∅)})
281	279, 280	mp1i 14	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩}:{∅}⟶{(𝑛‘∅)})
282		fvsng 7208	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((∅ ∈ V ∧ (𝑛‘∅) ∈ ℕ0) → ({⟨∅, (𝑛‘∅)⟩}‘∅) = (𝑛‘∅))
283	147, 52, 282	sylancr 599	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ({⟨∅, (𝑛‘∅)⟩}‘∅) = (𝑛‘∅))
284	283	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛‘∅) = ({⟨∅, (𝑛‘∅)⟩}‘∅))
285	147	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ∅ ∈ V)
286	145	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {∅} = 1o)
287	51, 52	fsnd 6892	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩}:{∅}⟶ℕ0)
288	286, 287	feq2dd 6717	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩}:1o⟶ℕ0)
289	288	ffnd 6732	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → {⟨∅, (𝑛‘∅)⟩} Fn 1o)
290	285, 144, 248, 289	fsneq 7057	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛 = {⟨∅, (𝑛‘∅)⟩} ↔ (𝑛‘∅) = ({⟨∅, (𝑛‘∅)⟩}‘∅)))
291	284, 290	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛 = {⟨∅, (𝑛‘∅)⟩})
292	144	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 1o = {∅})
293	291, 292	feq12d 6719	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑛:1o⟶{(𝑛‘∅)} ↔ {⟨∅, (𝑛‘∅)⟩}:{∅}⟶{(𝑛‘∅)}))
294	281, 293	mpbird 260	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → 𝑛:1o⟶{(𝑛‘∅)})
295	294	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑛:1o⟶{(𝑛‘∅)})
296	144	fneq2i 6659	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 Fn 1o ↔ 𝑖 Fn {∅})
297	250, 296	sylib 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖 Fn {∅})
298		0zd 12732	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 0 ∈ ℤ)
299	133	nn0zd 12745	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛‘∅) ∈ ℤ)
300	98	nn0zd 12745	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ∈ ℤ)
301	98	nn0ge0d 12697	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 0 ≤ (𝑖‘∅))
302	298, 299, 300, 301, 119	elfzd 13674	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑖‘∅) ∈ (0...(𝑛‘∅)))
303		fveq2 6908	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑜 = ∅ → (𝑖‘𝑜) = (𝑖‘∅))
304	303	eleq1d 2849	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑜 = ∅ → ((𝑖‘𝑜) ∈ (0...(𝑛‘∅)) ↔ (𝑖‘∅) ∈ (0...(𝑛‘∅))))
305	147, 304	ralsn 4642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑜 ∈ {∅} (𝑖‘𝑜) ∈ (0...(𝑛‘∅)) ↔ (𝑖‘∅) ∈ (0...(𝑛‘∅)))
306	302, 305	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ∀𝑜 ∈ {∅} (𝑖‘𝑜) ∈ (0...(𝑛‘∅)))
307		ffnfv 7143	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖:{∅}⟶(0...(𝑛‘∅)) ↔ (𝑖 Fn {∅} ∧ ∀𝑜 ∈ {∅} (𝑖‘𝑜) ∈ (0...(𝑛‘∅))))
308	297, 306, 307	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 𝑖:{∅}⟶(0...(𝑛‘∅)))
309	113	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → 1o ∈ V)
310	144, 309	eqeltrrid 2869	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → {∅} ∈ V)
311	144	ineq2i 4117	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (1o ∩ 1o) = (1o ∩ {∅})
312	311, 115	eqtr3i 2788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (1o ∩ {∅}) = 1o
313	277, 295, 308, 309, 310, 312	off 7735	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛 ∘f − 𝑖):1o⟶(0...(𝑛‘∅)))
314		fz0ssnn0 13783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (0...(𝑛‘∅)) ⊆ ℕ0
315	314	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (0...(𝑛‘∅)) ⊆ ℕ0)
316	313, 315	fssd 6749	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛 ∘f − 𝑖):1o⟶ℕ0)
317	316	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑛 ∘f − 𝑖):1o⟶ℕ0)
318	317, 247	ffvelcdmd 7109	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ((𝑛 ∘f − 𝑖)‘∅) ∈ ℕ0)
319	246, 318	eqeltrd 2864	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝑚‘∅) ∈ ℕ0)
320	258, 319	fsnd 6892	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑚‘∅)⟩}:{𝑋}⟶ℕ0)
321	320	ffnd 6732	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑚‘∅)⟩} Fn {𝑋})
322	263, 264, 265, 265, 124	offn 7730	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}) Fn {𝑋})
323	258, 204, 321, 322	fsneq 7057	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → ({⟨𝑋, (𝑚‘∅)⟩} = ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}) ↔ ({⟨𝑋, (𝑚‘∅)⟩}‘𝑋) = (({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})‘𝑋)))
324	270, 323	mpbird 260	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → {⟨𝑋, (𝑚‘∅)⟩} = ({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))
325	324	fveq2d 6912	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) ∧ 𝑚 = (𝑛 ∘f − 𝑖)) → (𝐺‘{⟨𝑋, (𝑚‘∅)⟩}) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
326	90, 309, 316	elmapdd 8889	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝑛 ∘f − 𝑖) ∈ (ℕ0 ↑m 1o))
327		fvexd 6923	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})) ∈ V)
328	244, 325, 326, 327	fvmptd 7024	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → ((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)) = (𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))
329	233, 328	oveq12d 7461	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛}) → (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))) = ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))
330	329	mpteq2dva 5201	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)))) = (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩})))))
331	330	oveq2d 7459	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ ((𝐹‘{⟨𝑋, (𝑖‘∅)⟩})(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − {⟨𝑋, (𝑖‘∅)⟩}))))))
332	216, 331	eqtr4d 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ {⟨𝑋, (𝑛‘∅)⟩}} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘({⟨𝑋, (𝑛‘∅)⟩} ∘f − 𝑗))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))))
333	19, 332	sylan9eqr 2820	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) ∧ 𝑚 = {⟨𝑋, (𝑛‘∅)⟩}) → (𝑅 Σg (𝑗 ∈ {𝑙 ∈ {𝑔 ∈ (ℕ0 ↑m {𝑋}) ∣ 𝑔 finSupp 0} ∣ 𝑙 ∘r ≤ 𝑚} ↦ ((𝐹‘𝑗)(.r‘𝑅)(𝐺‘(𝑚 ∘f − 𝑗))))) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))))
334		ovexd 7478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))) ∈ V)
335	13, 333, 60, 334	fvmptd 7024	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℕ0 ↑m 1o)) → ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩}) = (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖))))))
336	335	mpteq2dva 5201	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩})) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)))))))
337		eqid 2763	. . . . 5 ⊢ (1o mPoly 𝑅) = (1o mPoly 𝑅)
338		selvply1rhmlema.5	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝑅)
339		eqid 2763	. . . . . 6 ⊢ (Base‘𝑄) = (Base‘𝑄)
340	338, 339	ply1bas 22543	. . . . 5 ⊢ (Base‘𝑄) = (Base‘(1o mPoly 𝑅))
341		selvply1rhmlema.4	. . . . . 6 ⊢ × = (.r‘𝑄)
342	338, 337, 341	ply1mulr 22573	. . . . 5 ⊢ × = (.r‘(1o mPoly 𝑅))
343		psr1baslem 22533	. . . . 5 ⊢ (ℕ0 ↑m 1o) = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
344	5, 4, 7, 341, 338, 1, 46, 27, 10	selvply1rhmlema 34220	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐹) ∈ (Base‘𝑄))
345	5, 4, 7, 341, 338, 1, 46, 27, 11	selvply1rhmlema 34220	. . . . 5 ⊢ (𝜑 → (𝑀‘𝐺) ∈ (Base‘𝑄))
346	337, 340, 6, 342, 343, 344, 345	mplmul 22348	. . . 4 ⊢ (𝜑 → ((𝑀‘𝐹) × (𝑀‘𝐺)) = (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑅 Σg (𝑖 ∈ {𝑘 ∈ (ℕ0 ↑m 1o) ∣ 𝑘 ∘r ≤ 𝑛} ↦ (((𝑀‘𝐹)‘𝑖)(.r‘𝑅)((𝑀‘𝐺)‘(𝑛 ∘f − 𝑖)))))))
347	336, 346	eqtr4d 2801	. . 3 ⊢ (𝜑 → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ ((𝐹 · 𝐺)‘{⟨𝑋, (𝑛‘∅)⟩})) = ((𝑀‘𝐹) × (𝑀‘𝐺)))
348	3, 347	sylan9eqr 2820	. 2 ⊢ ((𝜑 ∧ 𝑓 = (𝐹 · 𝐺)) → (𝑛 ∈ (ℕ0 ↑m 1o) ↦ (𝑓‘{⟨𝑋, (𝑛‘∅)⟩})) = ((𝑀‘𝐹) × (𝑀‘𝐺)))
349	44	a1i 11	. . . 4 ⊢ (𝜑 → {𝑋} ∈ V)
350	4, 349, 27	mplringd 22360	. . 3 ⊢ (𝜑 → 𝑃 ∈ Ring)
351	5, 7, 350, 10, 11	ringcld 20514	. 2 ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)
352		ovexd 7478	. 2 ⊢ (𝜑 → ((𝑀‘𝐹) × (𝑀‘𝐺)) ∈ V)
353	1, 348, 351, 352	fvmptd2 7025	1 ⊢ (𝜑 → (𝑀‘(𝐹 · 𝐺)) = ((𝑀‘𝐹) × (𝑀‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  {csn 4588  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   Fn wfn 6553  ⟶wf 6554  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443   ∘_f cof 7715   ∘_r cofr 7716  1_oc1o 8497   ↑_m cmap 8875  Fincfn 9001   finSupp cfsupp 9381  0cc0 11227   ≤ cle 11371   − cmin 11568  ℕ₀cn0 12633  ...cfz 13666  Basecbs 17416  ._rcmulr 17458  0_gc0g 17639   Σ_g cgsu 17640  CMndccmn 20024  Ringcrg 20489   mPoly cmpl 22244  Poly₁cpl1 22525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-of 7717  df-ofr 7718  df-om 7902  df-1st 8025  df-2nd 8026  df-supp 8205  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-er 8745  df-map 8877  df-pm 8878  df-ixp 8954  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-fsupp 9382  df-sup 9461  df-oi 9531  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-5 12435  df-6 12436  df-7 12437  df-8 12438  df-9 12439  df-n0 12634  df-z 12721  df-dec 12842  df-uz 12993  df-fz 13667  df-fzo 13816  df-seq 14172  df-hash 14502  df-struct 17354  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-mulr 17471  df-sca 17473  df-vsca 17474  df-ip 17475  df-tset 17476  df-ple 17477  df-ds 17479  df-hom 17481  df-cco 17482  df-0g 17641  df-gsum 17642  df-prds 17647  df-pws 17649  df-mre 17785  df-mrc 17786  df-acs 17788  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-mhm 19007  df-submnd 19008  df-grp 19176  df-minusg 19177  df-mulg 19307  df-subg 19362  df-ghm 19457  df-cntz 19561  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-subrng 20828  df-subrg 20852  df-psr 22247  df-mpl 22249  df-opsr 22251  df-psr1 22528  df-ply1 22530

This theorem is used by:  selvply1rhm  34227