Theorem mplvrpmga 33710

Description: The action of permuting variables in a multivariate polynomial is a group action. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
mplvrpmga.1	⊢ 𝑆 = (SymGrp‘𝐼)
mplvrpmga.2	⊢ 𝑃 = (Base‘𝑆)
mplvrpmga.3	⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
mplvrpmga.4	⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
mplvrpmga.5	⊢ (𝜑 → 𝐼 ∈ 𝑉)

Assertion

Ref	Expression
mplvrpmga	⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐼,𝑑,𝑓,ℎ,𝑥   𝑀,𝑑,𝑓,𝑥   𝑃,𝑑,𝑓,𝑥   𝑥,𝑅   𝜑,𝑑,𝑓,𝑥

Allowed substitution hints:   𝜑(ℎ)   𝐴(ℎ)   𝑃(ℎ)   𝑅(𝑓,ℎ,𝑑)   𝑆(𝑥,𝑓,ℎ,𝑑)   𝑀(ℎ)   𝑉(𝑥,𝑓,ℎ,𝑑)

Proof of Theorem mplvrpmga

Dummy variables 𝑔 𝑝 𝑞 𝑐 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplvrpmga.5	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		mplvrpmga.1	. . . 4 ⊢ 𝑆 = (SymGrp‘𝐼)
3	2	symggrp 19329	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝑆 ∈ Grp)
4	1, 3	syl 17	. 2 ⊢ (𝜑 → 𝑆 ∈ Grp)
5		mplvrpmga.3	. . . 4 ⊢ 𝑀 = (Base‘(𝐼 mPoly 𝑅))
6	5	fvexi 6848	. . 3 ⊢ 𝑀 ∈ V
7	6	a1i 11	. 2 ⊢ (𝜑 → 𝑀 ∈ V)
8		fvexd 6849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (Base‘𝑅) ∈ V)
9		ovex 7391	. . . . . . . 8 ⊢ (ℕ0 ↑m 𝐼) ∈ V
10	9	rabex 5284	. . . . . . 7 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
12		eqid 2736	. . . . . . . . 9 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
13		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
14		eqid 2736	. . . . . . . . . 10 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
15	14	psrbasfsupp 33693	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
16		xp2nd 7966	. . . . . . . . . 10 ⊢ (𝑐 ∈ (𝑃 × 𝑀) → (2nd ‘𝑐) ∈ 𝑀)
17	16	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (2nd ‘𝑐) ∈ 𝑀)
18	12, 13, 5, 15, 17	mplelf 21953	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (2nd ‘𝑐):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
19		breq1 5101	. . . . . . . . 9 ⊢ (ℎ = (𝑥 ∘ (1st ‘𝑐)) → (ℎ finSupp 0 ↔ (𝑥 ∘ (1st ‘𝑐)) finSupp 0))
20		nn0ex 12407	. . . . . . . . . . 11 ⊢ ℕ0 ∈ V
21	20	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
22	1	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
23		ssrab2 4032	. . . . . . . . . . . . . 14 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
24	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
25	24	sselda 3933	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ (ℕ0 ↑m 𝐼))
26	22, 21, 25	elmaprd 32759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥:𝐼⟶ℕ0)
27		xp1st 7965	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ (𝑃 × 𝑀) → (1st ‘𝑐) ∈ 𝑃)
28	27	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1st ‘𝑐) ∈ 𝑃)
29		mplvrpmga.2	. . . . . . . . . . . . 13 ⊢ 𝑃 = (Base‘𝑆)
30	2, 29	symgbasf 19305	. . . . . . . . . . . 12 ⊢ ((1st ‘𝑐) ∈ 𝑃 → (1st ‘𝑐):𝐼⟶𝐼)
31	28, 30	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1st ‘𝑐):𝐼⟶𝐼)
32	26, 31	fcod 6687	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)):𝐼⟶ℕ0)
33	21, 22, 32	elmapdd 8778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)) ∈ (ℕ0 ↑m 𝐼))
34		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
35		breq1 5101	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑥 → (ℎ finSupp 0 ↔ 𝑥 finSupp 0))
36	35	elrab 3646	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝑥 ∈ (ℕ0 ↑m 𝐼) ∧ 𝑥 finSupp 0))
37	34, 36	sylib 218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∈ (ℕ0 ↑m 𝐼) ∧ 𝑥 finSupp 0))
38	37	simprd 495	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 finSupp 0)
39	2, 29	symgbasf1o 19304	. . . . . . . . . . 11 ⊢ ((1st ‘𝑐) ∈ 𝑃 → (1st ‘𝑐):𝐼–1-1-onto→𝐼)
40		f1of1 6773	. . . . . . . . . . 11 ⊢ ((1st ‘𝑐):𝐼–1-1-onto→𝐼 → (1st ‘𝑐):𝐼–1-1→𝐼)
41	28, 39, 40	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1st ‘𝑐):𝐼–1-1→𝐼)
42		0nn0 12416	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
43	42	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
44	38, 41, 43, 34	fsuppco 9305	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)) finSupp 0)
45	19, 33, 44	elrabd 3648	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ (1st ‘𝑐)) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
46	18, 45	ffvelcdmd 7030	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))) ∈ (Base‘𝑅))
47	46	fmpttd 7060	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
48	8, 11, 47	elmapdd 8778	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
49		eqid 2736	. . . . . . 7 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
50		eqid 2736	. . . . . . 7 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
51	49, 13, 15, 50, 1	psrbas 21889	. . . . . 6 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
52	51	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
53	48, 52	eleqtrrd 2839	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ (Base‘(𝐼 mPwSer 𝑅)))
54		coeq1 5806	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∘ (1st ‘𝑐)) = (𝑦 ∘ (1st ‘𝑐)))
55	54	fveq2d 6838	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))) = ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐))))
56	55	cbvmptv 5202	. . . . 5 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐))))
57		fveq1 6833	. . . . . . . 8 ⊢ (𝑔 = (2nd ‘𝑐) → (𝑔‘(𝑦 ∘ 𝑞)) = ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞)))
58	57	mpteq2dv 5192	. . . . . . 7 ⊢ (𝑔 = (2nd ‘𝑐) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))))
59	58	breq1d 5108	. . . . . 6 ⊢ (𝑔 = (2nd ‘𝑐) → ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅) ↔ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅)))
60		coeq2 5807	. . . . . . . . 9 ⊢ (𝑞 = (1st ‘𝑐) → (𝑦 ∘ 𝑞) = (𝑦 ∘ (1st ‘𝑐)))
61	60	fveq2d 6838	. . . . . . . 8 ⊢ (𝑞 = (1st ‘𝑐) → ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞)) = ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐))))
62	61	mpteq2dv 5192	. . . . . . 7 ⊢ (𝑞 = (1st ‘𝑐) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐)))))
63	62	breq1d 5108	. . . . . 6 ⊢ (𝑞 = (1st ‘𝑐) → ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅) ↔ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅)))
64		mplvrpmga.4	. . . . . . . . . . . . 13 ⊢ 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
65	64	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
66		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → 𝑓 = 𝑔)
67		coeq2 5807	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 = 𝑞 → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑞))
68	67	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑞))
69	66, 68	fveq12d 6841	. . . . . . . . . . . . . 14 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘(𝑥 ∘ 𝑞)))
70	69	mpteq2dv 5192	. . . . . . . . . . . . 13 ⊢ ((𝑑 = 𝑞 ∧ 𝑓 = 𝑔) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))))
71	70	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) ∧ (𝑑 = 𝑞 ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))))
72		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝑞 ∈ 𝑃)
73		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝑔 ∈ 𝑀)
74	10	mptex 7169	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))) ∈ V
75	74	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))) ∈ V)
76	65, 71, 72, 73, 75	ovmpod 7510	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))))
77		coeq1 5806	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 ∘ 𝑞) = (𝑦 ∘ 𝑞))
78	77	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑔‘(𝑥 ∘ 𝑞)) = (𝑔‘(𝑦 ∘ 𝑞)))
79	78	cbvmptv 5202	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ 𝑞))) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))
80	76, 79	eqtrdi 2787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
81	1	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → 𝐼 ∈ 𝑉)
82		eqid 2736	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
83	2, 29, 5, 64, 81, 82, 73, 72	mplvrpmfgalem 33709	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) finSupp (0g‘𝑅))
84	80, 83	eqbrtrrd 5122	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
85	84	anasss 466	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑀 ∧ 𝑞 ∈ 𝑃)) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
86	85	ralrimivva 3179	. . . . . . 7 ⊢ (𝜑 → ∀𝑔 ∈ 𝑀 ∀𝑞 ∈ 𝑃 (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
87	86	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → ∀𝑔 ∈ 𝑀 ∀𝑞 ∈ 𝑃 (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
88	16	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (2nd ‘𝑐) ∈ 𝑀)
89	27	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (1st ‘𝑐) ∈ 𝑃)
90	59, 63, 87, 88, 89	rspc2dv 3591	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑦 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅))
91	56, 90	eqbrtrid 5133	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅))
92	12, 49, 50, 82, 5	mplelbas 21946	. . . 4 ⊢ ((𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ 𝑀 ↔ ((𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) finSupp (0g‘𝑅)))
93	53, 91, 92	sylanbrc 583	. . 3 ⊢ ((𝜑 ∧ 𝑐 ∈ (𝑃 × 𝑀)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) ∈ 𝑀)
94		vex 3444	. . . . . . . 8 ⊢ 𝑑 ∈ V
95		vex 3444	. . . . . . . 8 ⊢ 𝑓 ∈ V
96	94, 95	op2ndd 7944	. . . . . . 7 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (2nd ‘𝑐) = 𝑓)
97	94, 95	op1std 7943	. . . . . . . 8 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (1st ‘𝑐) = 𝑑)
98	97	coeq2d 5811	. . . . . . 7 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (𝑥 ∘ (1st ‘𝑐)) = (𝑥 ∘ 𝑑))
99	96, 98	fveq12d 6841	. . . . . 6 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))) = (𝑓‘(𝑥 ∘ 𝑑)))
100	99	mpteq2dv 5192	. . . . 5 ⊢ (𝑐 = ⟨𝑑, 𝑓⟩ → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
101	100	mpompt 7472	. . . 4 ⊢ (𝑐 ∈ (𝑃 × 𝑀) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐))))) = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))))
102	64, 101	eqtr4i 2762	. . 3 ⊢ 𝐴 = (𝑐 ∈ (𝑃 × 𝑀) ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((2nd ‘𝑐)‘(𝑥 ∘ (1st ‘𝑐)))))
103	93, 102	fmptd 7059	. 2 ⊢ (𝜑 → 𝐴:(𝑃 × 𝑀)⟶𝑀)
104	2	symgid 19330	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑉 → ( I ↾ 𝐼) = (0g‘𝑆))
105	1, 104	syl 17	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐼) = (0g‘𝑆))
106	105	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ( I ↾ 𝐼) = (0g‘𝑆))
107	106	oveq1d 7373	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (( I ↾ 𝐼)𝐴𝑔) = ((0g‘𝑆)𝐴𝑔))
108	64	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
109	1	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
110	20	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
111	23	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
112	111	sselda 3933	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ (ℕ0 ↑m 𝐼))
113	109, 110, 112	elmaprd 32759	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥:𝐼⟶ℕ0)
114		fcoi1 6708	. . . . . . . . . . 11 ⊢ (𝑥:𝐼⟶ℕ0 → (𝑥 ∘ ( I ↾ 𝐼)) = 𝑥)
115	113, 114	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ ( I ↾ 𝐼)) = 𝑥)
116	115	fveq2d 6838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘(𝑥 ∘ ( I ↾ 𝐼))) = (𝑔‘𝑥))
117	116	mpteq2dva 5191	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
118	117	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
119		simpr 484	. . . . . . . . . 10 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → 𝑓 = 𝑔)
120		coeq2 5807	. . . . . . . . . . 11 ⊢ (𝑑 = ( I ↾ 𝐼) → (𝑥 ∘ 𝑑) = (𝑥 ∘ ( I ↾ 𝐼)))
121	120	adantr 480	. . . . . . . . . 10 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → (𝑥 ∘ 𝑑) = (𝑥 ∘ ( I ↾ 𝐼)))
122	119, 121	fveq12d 6841	. . . . . . . . 9 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘(𝑥 ∘ ( I ↾ 𝐼))))
123	122	mpteq2dv 5192	. . . . . . . 8 ⊢ ((𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))))
124	123	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ ( I ↾ 𝐼)))))
125	12, 49, 50, 82, 5	mplelbas 21946	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ 𝑀 ↔ (𝑔 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑔 finSupp (0g‘𝑅)))
126	125	simplbi 497	. . . . . . . . . . 11 ⊢ (𝑔 ∈ 𝑀 → 𝑔 ∈ (Base‘(𝐼 mPwSer 𝑅)))
127	49, 13, 15, 50, 126	psrelbas 21890	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑀 → 𝑔:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
128	127	ad3antlr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑑 = ( I ↾ 𝐼)) ∧ 𝑓 = 𝑔) → 𝑔:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
129	128	feqmptd 6902	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑑 = ( I ↾ 𝐼)) ∧ 𝑓 = 𝑔) → 𝑔 = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
130	129	anasss 466	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → 𝑔 = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘𝑥)))
131	118, 124, 130	3eqtr4d 2781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑑 = ( I ↾ 𝐼) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = 𝑔)
132		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
133	29, 132	grpidcl 18895	. . . . . . . . 9 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝑃)
134	1, 3, 133	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝑃)
135	105, 134	eqeltrd 2836	. . . . . . 7 ⊢ (𝜑 → ( I ↾ 𝐼) ∈ 𝑃)
136	135	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ( I ↾ 𝐼) ∈ 𝑃)
137		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → 𝑔 ∈ 𝑀)
138	108, 131, 136, 137, 137	ovmpod 7510	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (( I ↾ 𝐼)𝐴𝑔) = 𝑔)
139	107, 138	eqtr3d 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ((0g‘𝑆)𝐴𝑔) = 𝑔)
140		eqid 2736	. . . . . . . . . 10 ⊢ (+g‘𝑆) = (+g‘𝑆)
141	2, 29, 140	symgov 19313	. . . . . . . . 9 ⊢ ((𝑝 ∈ 𝑃 ∧ 𝑞 ∈ 𝑃) → (𝑝(+g‘𝑆)𝑞) = (𝑝 ∘ 𝑞))
142	141	adantll 714	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝(+g‘𝑆)𝑞) = (𝑝 ∘ 𝑞))
143	142	oveq1d 7373	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = ((𝑝 ∘ 𝑞)𝐴𝑔))
144		coass 6224	. . . . . . . . . . 11 ⊢ ((𝑥 ∘ 𝑝) ∘ 𝑞) = (𝑥 ∘ (𝑝 ∘ 𝑞))
145	144	a1i 11	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((𝑥 ∘ 𝑝) ∘ 𝑞) = (𝑥 ∘ (𝑝 ∘ 𝑞)))
146	145	fveq2d 6838	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)) = (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞))))
147	146	mpteq2dva 5191	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
148	80	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑞𝐴𝑔) = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
149	148	oveq2d 7374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝𝐴(𝑞𝐴𝑔)) = (𝑝𝐴(𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))))
150	64	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝐴 = (𝑑 ∈ 𝑃, 𝑓 ∈ 𝑀 ↦ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑)))))
151		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑑 = 𝑝)
152	151	coeq2d 5811	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑑) = (𝑥 ∘ 𝑝))
153	152	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑓‘(𝑥 ∘ 𝑝)))
154		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
155		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑦 = (𝑥 ∘ 𝑝)) → 𝑦 = (𝑥 ∘ 𝑝))
156	155	coeq1d 5810	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑦 = (𝑥 ∘ 𝑝)) → (𝑦 ∘ 𝑞) = ((𝑥 ∘ 𝑝) ∘ 𝑞))
157	156	fveq2d 6838	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑦 = (𝑥 ∘ 𝑝)) → (𝑔‘(𝑦 ∘ 𝑞)) = (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)))
158		breq1 5101	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑥 ∘ 𝑝) → (ℎ finSupp 0 ↔ (𝑥 ∘ 𝑝) finSupp 0))
159	20	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
160	1	ad3antrrr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝐼 ∈ 𝑉)
161	160	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
162	23	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
163	162	sselda 3933	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ (ℕ0 ↑m 𝐼))
164	161, 159, 163	elmaprd 32759	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥:𝐼⟶ℕ0)
165	2, 29	symgbasf 19305	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑃 → 𝑝:𝐼⟶𝐼)
166	165	ad5antlr 735	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑝:𝐼⟶𝐼)
167	164, 166	fcod 6687	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝):𝐼⟶ℕ0)
168	159, 161, 167	elmapdd 8778	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝) ∈ (ℕ0 ↑m 𝐼))
169	36	simprbi 496	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} → 𝑥 finSupp 0)
170	169	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 finSupp 0)
171	2, 29	symgbasf1o 19304	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ 𝑃 → 𝑝:𝐼–1-1-onto→𝐼)
172		f1of1 6773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝:𝐼–1-1-onto→𝐼 → 𝑝:𝐼–1-1→𝐼)
173	171, 172	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ 𝑃 → 𝑝:𝐼–1-1→𝐼)
174	173	ad5antlr 735	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑝:𝐼–1-1→𝐼)
175	42	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
176		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
177	170, 174, 175, 176	fsuppco 9305	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝) finSupp 0)
178	158, 168, 177	elrabd 3648	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑥 ∘ 𝑝) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
179		fvexd 6849	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)) ∈ V)
180		nfv 1915	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝)
181		nfmpt1 5197	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦(𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))
182	181	nfeq2 2916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))
183	180, 182	nfan 1900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))
184		nfv 1915	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
185	183, 184	nfan 1900	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
186		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑥 ∘ 𝑝)
187		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))
188	154, 157, 178, 179, 185, 186, 187	fvmptdf 6947	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑓‘(𝑥 ∘ 𝑝)) = (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)))
189	153, 188	eqtrd 2771	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) ∧ 𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞)))
190	189	mpteq2dva 5191	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑑 = 𝑝) ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
191	190	anasss 466	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ (𝑑 = 𝑝 ∧ 𝑓 = (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))))) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
192		simplr 768	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑝 ∈ 𝑃)
193		fvexd 6849	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (Base‘𝑅) ∈ V)
194	10	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
195	137	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑔 ∈ 𝑀)
196	12, 13, 5, 15, 195	mplelf 21953	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑔:{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
197		breq1 5101	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑦 ∘ 𝑞) → (ℎ finSupp 0 ↔ (𝑦 ∘ 𝑞) finSupp 0))
198	20	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
199	160	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
200	23	a1i 11	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
201	200	sselda 3933	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦 ∈ (ℕ0 ↑m 𝐼))
202	199, 198, 201	elmaprd 32759	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦:𝐼⟶ℕ0)
203	2, 29	symgbasf 19305	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ 𝑃 → 𝑞:𝐼⟶𝐼)
204	203	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑞:𝐼⟶𝐼)
205	202, 204	fcod 6687	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞):𝐼⟶ℕ0)
206	198, 199, 205	elmapdd 8778	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞) ∈ (ℕ0 ↑m 𝐼))
207		breq1 5101	. . . . . . . . . . . . . . . . . . . 20 ⊢ (ℎ = 𝑦 → (ℎ finSupp 0 ↔ 𝑦 finSupp 0))
208	207	elrab 3646	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↔ (𝑦 ∈ (ℕ0 ↑m 𝐼) ∧ 𝑦 finSupp 0))
209	208	simprbi 496	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} → 𝑦 finSupp 0)
210	209	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦 finSupp 0)
211	2, 29	symgbasf1o 19304	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑞 ∈ 𝑃 → 𝑞:𝐼–1-1-onto→𝐼)
212	211	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑞:𝐼–1-1-onto→𝐼)
213		f1of1 6773	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞:𝐼–1-1-onto→𝐼 → 𝑞:𝐼–1-1→𝐼)
214	212, 213	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑞:𝐼–1-1→𝐼)
215	42	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
216		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
217	210, 214, 215, 216	fsuppco 9305	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞) finSupp 0)
218	197, 206, 217	elrabd 3648	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑦 ∘ 𝑞) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
219	196, 218	ffvelcdmd 7030	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ 𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑔‘(𝑦 ∘ 𝑞)) ∈ (Base‘𝑅))
220	219	fmpttd 7060	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶(Base‘𝑅))
221	193, 194, 220	elmapdd 8778	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
222	51	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (Base‘(𝐼 mPwSer 𝑅)) = ((Base‘𝑅) ↑m {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}))
223	221, 222	eleqtrrd 2839	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ (Base‘(𝐼 mPwSer 𝑅)))
224	84	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅))
225	12, 49, 50, 82, 5	mplelbas 21946	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ 𝑀 ↔ ((𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) finSupp (0g‘𝑅)))
226	223, 224, 225	sylanbrc 583	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞))) ∈ 𝑀)
227	194	mptexd 7170	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))) ∈ V)
228	150, 191, 192, 226, 227	ovmpod 7510	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝𝐴(𝑦 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑦 ∘ 𝑞)))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
229	149, 228	eqtrd 2771	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝𝐴(𝑞𝐴𝑔)) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘((𝑥 ∘ 𝑝) ∘ 𝑞))))
230		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → 𝑓 = 𝑔)
231		coeq2 5807	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑝 ∘ 𝑞) → (𝑥 ∘ 𝑑) = (𝑥 ∘ (𝑝 ∘ 𝑞)))
232	231	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → (𝑥 ∘ 𝑑) = (𝑥 ∘ (𝑝 ∘ 𝑞)))
233	230, 232	fveq12d 6841	. . . . . . . . . . 11 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → (𝑓‘(𝑥 ∘ 𝑑)) = (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞))))
234	233	mpteq2dv 5192	. . . . . . . . . 10 ⊢ ((𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
235	234	adantl 481	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) ∧ (𝑑 = (𝑝 ∘ 𝑞) ∧ 𝑓 = 𝑔)) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑓‘(𝑥 ∘ 𝑑))) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
236	160, 3	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑆 ∈ Grp)
237		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑞 ∈ 𝑃)
238	29, 140, 236, 192, 237	grpcld 18877	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝(+g‘𝑆)𝑞) ∈ 𝑃)
239	142, 238	eqeltrrd 2837	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑝 ∘ 𝑞) ∈ 𝑃)
240		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → 𝑔 ∈ 𝑀)
241	194	mptexd 7170	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))) ∈ V)
242	150, 235, 239, 240, 241	ovmpod 7510	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝 ∘ 𝑞)𝐴𝑔) = (𝑥 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ (𝑔‘(𝑥 ∘ (𝑝 ∘ 𝑞)))))
243	147, 229, 242	3eqtr4rd 2782	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝 ∘ 𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
244	143, 243	eqtrd 2771	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ 𝑝 ∈ 𝑃) ∧ 𝑞 ∈ 𝑃) → ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
245	244	anasss 466	. . . . 5 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑀) ∧ (𝑝 ∈ 𝑃 ∧ 𝑞 ∈ 𝑃)) → ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
246	245	ralrimivva 3179	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔)))
247	139, 246	jca 511	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑀) → (((0g‘𝑆)𝐴𝑔) = 𝑔 ∧ ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔))))
248	247	ralrimiva 3128	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑀 (((0g‘𝑆)𝐴𝑔) = 𝑔 ∧ ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔))))
249	29, 140, 132	isga 19220	. 2 ⊢ (𝐴 ∈ (𝑆 GrpAct 𝑀) ↔ ((𝑆 ∈ Grp ∧ 𝑀 ∈ V) ∧ (𝐴:(𝑃 × 𝑀)⟶𝑀 ∧ ∀𝑔 ∈ 𝑀 (((0g‘𝑆)𝐴𝑔) = 𝑔 ∧ ∀𝑝 ∈ 𝑃 ∀𝑞 ∈ 𝑃 ((𝑝(+g‘𝑆)𝑞)𝐴𝑔) = (𝑝𝐴(𝑞𝐴𝑔))))))
250	4, 7, 103, 248, 249	syl22anbrc 32529	1 ⊢ (𝜑 → 𝐴 ∈ (𝑆 GrpAct 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  {crab 3399  Vcvv 3440   ⊆ wss 3901  ⟨cop 4586   class class class wbr 5098   ↦ cmpt 5179   I cid 5518   × cxp 5622   ↾ cres 5626   ∘ ccom 5628  ⟶wf 6488  –1-1→wf1 6489  –1-1-onto→wf1o 6491  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  1^st c1st 7931  2^nd c2nd 7932   ↑_m cmap 8763   finSupp cfsupp 9264  0cc0 11026  ℕ₀cn0 12401  Basecbs 17136  +_gcplusg 17177  0_gc0g 17359  Grpcgrp 18863   GrpAct cga 19218  SymGrpcsymg 19298   mPwSer cmps 21860   mPoly cmpl 21862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-tset 17196  df-0g 17361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18709  df-efmnd 18794  df-grp 18866  df-ga 19219  df-symg 19299  df-psr 21865  df-mpl 21867

This theorem is referenced by:  mplvrpmmhm  33711  mplvrpmrhm  33712  splysubrg  33718  issply  33719