Theorem evlextv 33873

Description: Evaluating a variable-extended polynomial is the same as evaluating the polynomial in the original set of variables (in both cases, the additionial variable is ignored). (Contributed by Thierry Arnoux, 15-Feb-2026.)

Hypotheses

Ref	Expression
evlextv.q	⊢ 𝑄 = (𝐼 eval 𝑅)
evlextv.o	⊢ 𝑂 = (𝐽 eval 𝑅)
evlextv.j	⊢ 𝐽 = (𝐼 ∖ {𝑌})
evlextv.m	⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
evlextv.b	⊢ 𝐵 = (Base‘𝑅)
evlextv.e	⊢ 𝐸 = (𝐼extendVars𝑅)
evlextv.r	⊢ (𝜑 → 𝑅 ∈ CRing)
evlextv.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
evlextv.y	⊢ (𝜑 → 𝑌 ∈ 𝐼)
evlextv.f	⊢ (𝜑 → 𝐹 ∈ 𝑀)
evlextv.a	⊢ (𝜑 → 𝐴:𝐼⟶𝐵)

Assertion

Ref	Expression
evlextv	⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)))

Proof of Theorem evlextv

Dummy variables 𝑏 𝑐 𝑖 𝑥 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlextv.e	. . . . . . . . . . 11 ⊢ 𝐸 = (𝐼extendVars𝑅)
2	1	fveq1i 6880	. . . . . . . . . 10 ⊢ (𝐸‘𝑌) = ((𝐼extendVars𝑅)‘𝑌)
3	2	fveq1i 6880	. . . . . . . . 9 ⊢ ((𝐸‘𝑌)‘𝐹) = (((𝐼extendVars𝑅)‘𝑌)‘𝐹)
4	3	fveq1i 6880	. . . . . . . 8 ⊢ (((𝐸‘𝑌)‘𝐹)‘𝑐) = ((((𝐼extendVars𝑅)‘𝑌)‘𝐹)‘𝑐)
5	4	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (((𝐸‘𝑌)‘𝐹)‘𝑐) = ((((𝐼extendVars𝑅)‘𝑌)‘𝐹)‘𝑐))
6		eqid 2769	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
7		eqid 2769	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
8		evlextv.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
9	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝐼 ∈ 𝑉)
10		evlextv.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ CRing)
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑅 ∈ CRing)
12		evlextv.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐼)
13	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑌 ∈ 𝐼)
14		evlextv.j	. . . . . . . 8 ⊢ 𝐽 = (𝐼 ∖ {𝑌})
15		evlextv.m	. . . . . . . 8 ⊢ 𝑀 = (Base‘(𝐽 mPoly 𝑅))
16		evlextv.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ 𝑀)
17	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝐹 ∈ 𝑀)
18		breq1 5113	. . . . . . . . 9 ⊢ (ℎ = 𝑐 → (ℎ finSupp 0 ↔ 𝑐 finSupp 0))
19		ssrab2 4042	. . . . . . . . . . 11 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ⊆ (ℕ0 ↑m 𝐼)
20	19	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ⊆ (ℕ0 ↑m 𝐼))
21	20	sselda 3945	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐 ∈ (ℕ0 ↑m 𝐼))
22		fveq1 6878	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑐 → (ℎ‘𝑌) = (𝑐‘𝑌))
23	22	eqeq1d 2771	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑐 → ((ℎ‘𝑌) = 0 ↔ (𝑐‘𝑌) = 0))
24	18, 23	anbi12d 643	. . . . . . . . . . 11 ⊢ (ℎ = 𝑐 → ((ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0) ↔ (𝑐 finSupp 0 ∧ (𝑐‘𝑌) = 0)))
25		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})
26	24, 25	elrabrd 3662	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 finSupp 0 ∧ (𝑐‘𝑌) = 0))
27	26	simpld 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐 finSupp 0)
28	18, 21, 27	elrabd 3661	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
29	6, 7, 9, 11, 13, 14, 15, 17, 28	extvfvv 33865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → ((((𝐼extendVars𝑅)‘𝑌)‘𝐹)‘𝑐) = if((𝑐‘𝑌) = 0, (𝐹‘(𝑐 ↾ 𝐽)), (0g‘𝑅)))
30	26	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐‘𝑌) = 0)
31	30	iftrued 4497	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → if((𝑐‘𝑌) = 0, (𝐹‘(𝑐 ↾ 𝐽)), (0g‘𝑅)) = (𝐹‘(𝑐 ↾ 𝐽)))
32	5, 29, 31	3eqtrd 2808	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (((𝐸‘𝑌)‘𝐹)‘𝑐) = (𝐹‘(𝑐 ↾ 𝐽)))
33		eqid 2769	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
34		evlextv.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
35	33, 34	mgpbas 20217	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
36		eqid 2769	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
37	33, 36	ringidval 20261	. . . . . . . 8 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
38	33	crngmgp 20319	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
39	11, 38	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (mulGrp‘𝑅) ∈ CMnd)
40		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 ∈ (𝐼 ∖ 𝐽))
41	14	difeq2i 4086	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∖ 𝐽) = (𝐼 ∖ (𝐼 ∖ {𝑌}))
42	12	snssd 4754	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {𝑌} ⊆ 𝐼)
43		dfss4 4230	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑌} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑌})) = {𝑌})
44	42, 43	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ∖ (𝐼 ∖ {𝑌})) = {𝑌})
45	41, 44	eqtrid 2816	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ∖ 𝐽) = {𝑌})
46	45	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝐼 ∖ 𝐽) = {𝑌})
47	40, 46	eleqtrd 2871	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 ∈ {𝑌})
48	47	elsnd 4609	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 = 𝑌)
49	48	fveq2d 6883	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑖) = (𝑐‘𝑌))
50	30	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑌) = 0)
51	49, 50	eqtrd 2804	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑖) = 0)
52	51	oveq1d 7423	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = (0(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))
53		evlextv.a	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)
54	53	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝐴:𝐼⟶𝐵)
55		difssd 4099	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
56	55	sselda 3945	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 ∈ 𝐼)
57	54, 56	ffvelcdmd 7078	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝐴‘𝑖) ∈ 𝐵)
58		eqid 2769	. . . . . . . . . . 11 ⊢ (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
59	35, 37, 58	mulg0 19136	. . . . . . . . . 10 ⊢ ((𝐴‘𝑖) ∈ 𝐵 → (0(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = (1r‘𝑅))
60	57, 59	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (0(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = (1r‘𝑅))
61	52, 60	eqtrd 2804	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = (1r‘𝑅))
62		fvexd 6894	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1r‘𝑅) ∈ V)
63		0nn0 12515	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
65	8	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
66		ssidd 3968	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ⊆ 𝐼)
67	53	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐴:𝐼⟶𝐵)
68	67	ffvelcdmda 7077	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) ∈ 𝐵)
69		nn0ex 12506	. . . . . . . . . . . 12 ⊢ ℕ0 ∈ V
70	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
71		ssrab2 4042	. . . . . . . . . . . . 13 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼)
72	71	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐼))
73	72	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐 ∈ (ℕ0 ↑m 𝐼))
74	65, 70, 73	elmaprd 32962	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐:𝐼⟶ℕ0)
75		simpr 489	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
76	18, 75	elrabrd 3662	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐 finSupp 0)
77	35, 37, 58	mulg0 19136	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (0(.g‘(mulGrp‘𝑅))𝑥) = (1r‘𝑅))
78	77	adantl 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑥 ∈ 𝐵) → (0(.g‘(mulGrp‘𝑅))𝑥) = (1r‘𝑅))
79	62, 64, 65, 66, 68, 74, 76, 78	fisuppov1 32965	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) finSupp (1r‘𝑅))
80	28, 79	syldan 602	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))) finSupp (1r‘𝑅))
81	10, 38	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
82	81	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ CMnd)
83	82	cmnmndd 19870	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ Mnd)
84	83	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑅) ∈ Mnd)
85	74	ffvelcdmda 7077	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → (𝑐‘𝑖) ∈ ℕ0)
86	35, 58, 84, 85, 68	mulgnn0cld 19157	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) ∈ 𝐵)
87	28, 86	syldanl 613	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐼) → ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) ∈ 𝐵)
88		difss 4098	. . . . . . . . . 10 ⊢ (𝐼 ∖ {𝑌}) ⊆ 𝐼
89	14, 88	eqsstri 3991	. . . . . . . . 9 ⊢ 𝐽 ⊆ 𝐼
90	89	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝐽 ⊆ 𝐼)
91	35, 37, 39, 9, 61, 80, 87, 90	gsummptfsres 33311	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) = ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))
92		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → 𝑖 ∈ 𝐽)
93	92	fvresd 6899	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → ((𝑐 ↾ 𝐽)‘𝑖) = (𝑐‘𝑖))
94	92	fvresd 6899	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑖) = (𝐴‘𝑖))
95	93, 94	oveq12d 7426	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)) = ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))
96	95	mpteq2dva 5205	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))) = (𝑖 ∈ 𝐽 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))
97	96	oveq2d 7424	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))) = ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))
98	91, 97	eqtr4d 2807	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) = ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))))
99	32, 98	oveq12d 7426	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))) = ((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))
100	99	mpteq2dva 5205	. . . 4 ⊢ (𝜑 → (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))) = (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ↦ ((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))))))
101	100	oveq2d 7424	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))) = (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ↦ ((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))))
102	10	crngringd 20324	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
103	102	ringcmnd 20363	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
104		ovex 7441	. . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V
105	104	rabex 5307	. . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V
106	105	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∈ V)
107	4	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → (((𝐸‘𝑌)‘𝐹)‘𝑐) = ((((𝐼extendVars𝑅)‘𝑌)‘𝐹)‘𝑐))
108	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝐼 ∈ 𝑉)
109	10	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝑅 ∈ CRing)
110	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝑌 ∈ 𝐼)
111	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝐹 ∈ 𝑀)
112		difssd 4099	. . . . . . . . 9 ⊢ (𝜑 → ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ⊆ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
113	112	sselda 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
114	6, 7, 108, 109, 110, 14, 15, 111, 113	extvfvv 33865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → ((((𝐼extendVars𝑅)‘𝑌)‘𝐹)‘𝑐) = if((𝑐‘𝑌) = 0, (𝐹‘(𝑐 ↾ 𝐽)), (0g‘𝑅)))
115	113	adantr 485	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
116	71, 115	sselid 3943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → 𝑐 ∈ (ℕ0 ↑m 𝐼))
117	18, 115	elrabrd 3662	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → 𝑐 finSupp 0)
118		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → (𝑐‘𝑌) = 0)
119	117, 118	jca 520	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → (𝑐 finSupp 0 ∧ (𝑐‘𝑌) = 0))
120	24, 116, 119	elrabd 3661	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})
121		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}))
122	121	eldifbd 3926	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) ∧ (𝑐‘𝑌) = 0) → ¬ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})
123	120, 122	pm2.65da 828	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → ¬ (𝑐‘𝑌) = 0)
124	123	iffalsed 4500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → if((𝑐‘𝑌) = 0, (𝐹‘(𝑐 ↾ 𝐽)), (0g‘𝑅)) = (0g‘𝑅))
125	107, 114, 124	3eqtrd 2808	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → (((𝐸‘𝑌)‘𝐹)‘𝑐) = (0g‘𝑅))
126	125	oveq1d 7423	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))) = ((0g‘𝑅)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))
127		eqid 2769	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
128	102	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝑅 ∈ Ring)
129	86	fmpttd 7108	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))):𝐼⟶𝐵)
130	35, 37, 82, 65, 129, 79	gsumcl 19981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) ∈ 𝐵)
131	113, 130	syldan 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))) ∈ 𝐵)
132	34, 127, 7, 128, 131	ringlzd 20374	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → ((0g‘𝑅)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))) = (0g‘𝑅))
133	126, 132	eqtrd 2804	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))) = (0g‘𝑅))
134		eqid 2769	. . . . . 6 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
135		eqid 2769	. . . . . 6 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅))
136	6	psrbasfsupp 33842	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
137	6, 7, 8, 102, 34, 14, 15, 12, 16, 135	extvfvcl 33867	. . . . . . 7 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝑌)‘𝐹) ∈ (Base‘(𝐼 mPoly 𝑅)))
138	3, 137	eqeltrid 2873	. . . . . 6 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹) ∈ (Base‘(𝐼 mPoly 𝑅)))
139	134, 34, 135, 136, 138	mplelf 22112	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶𝐵)
140	134, 135, 7, 138	mplelsfi 22109	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹) finSupp (0g‘𝑅))
141	34, 102, 106, 130, 139, 140	rmfsupp2 33494	. . . 4 ⊢ (𝜑 → (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)))))) finSupp (0g‘𝑅))
142	102	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
143	139	ffvelcdmda 7077	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (((𝐸‘𝑌)‘𝐹)‘𝑐) ∈ 𝐵)
144	34, 127, 142, 143, 130	ringcld 20338	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))) ∈ 𝐵)
145		simpl 487	. . . . . 6 ⊢ ((ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0) → ℎ finSupp 0)
146	145	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ ℎ ∈ (ℕ0 ↑m 𝐼)) → ((ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0) → ℎ finSupp 0))
147	146	ss2rabdv 4037	. . . 4 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ⊆ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
148	34, 7, 103, 106, 133, 141, 144, 147	gsummptfsres 33311	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))) = (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))))
149		nfcv 2931	. . . 4 ⊢ Ⅎ𝑏((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))))
150		fveq2 6879	. . . . 5 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → (𝐹‘𝑏) = (𝐹‘(𝑐 ↾ 𝐽)))
151		fveq1 6878	. . . . . . . 8 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → (𝑏‘𝑖) = ((𝑐 ↾ 𝐽)‘𝑖))
152	151	oveq1d 7423	. . . . . . 7 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)) = (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))
153	152	mpteq2dv 5206	. . . . . 6 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))) = (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))
154	153	oveq2d 7424	. . . . 5 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))) = ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))))
155	150, 154	oveq12d 7426	. . . 4 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → ((𝐹‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))) = ((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))
156		ovex 7441	. . . . . 6 ⊢ (ℕ0 ↑m 𝐽) ∈ V
157	156	rabex 5307	. . . . 5 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ∈ V
158	157	a1i 11	. . . 4 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ∈ V)
159		eqid 2769	. . . . . . . 8 ⊢ (𝐽 mPoly 𝑅) = (𝐽 mPoly 𝑅)
160		eqid 2769	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}
161	160	psrbasfsupp 33842	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ (◡ℎ “ ℕ) ∈ Fin}
162	159, 34, 15, 161, 16	mplelf 22112	. . . . . . 7 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}⟶𝐵)
163	162	feqmptd 6947	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ (𝐹‘𝑏)))
164	159, 15, 7, 16	mplelsfi 22109	. . . . . 6 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
165	163, 164	eqbrtrrd 5136	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ (𝐹‘𝑏)) finSupp (0g‘𝑅))
166	102	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
167		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
168	34, 127, 7, 166, 167	ringlzd 20374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((0g‘𝑅)(.r‘𝑅)𝑥) = (0g‘𝑅))
169	162	ffvelcdmda 7077	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐹‘𝑏) ∈ 𝐵)
170	81	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ CMnd)
171	89	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ⊆ 𝐼)
172	8, 171	ssexd 5292	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
173	172	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝐽 ∈ V)
174	170	cmnmndd 19870	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ Mnd)
175	174	adantr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → (mulGrp‘𝑅) ∈ Mnd)
176	69	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
177		ssrab2 4042	. . . . . . . . . . . 12 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐽)
178	177	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ⊆ (ℕ0 ↑m 𝐽))
179	178	sselda 3945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 ∈ (ℕ0 ↑m 𝐽))
180	173, 176, 179	elmaprd 32962	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏:𝐽⟶ℕ0)
181	180	ffvelcdmda 7077	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → (𝑏‘𝑖) ∈ ℕ0)
182	53	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝐴:𝐼⟶𝐵)
183	89	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝐽 ⊆ 𝐼)
184	182, 183	fssresd 6743	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐴 ↾ 𝐽):𝐽⟶𝐵)
185	184	ffvelcdmda 7077	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑖) ∈ 𝐵)
186	35, 58, 175, 181, 185	mulgnn0cld 19157	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)) ∈ 𝐵)
187	186	fmpttd 7108	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))):𝐽⟶𝐵)
188	180	feqmptd 6947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 = (𝑖 ∈ 𝐽 ↦ (𝑏‘𝑖)))
189		breq1 5113	. . . . . . . . 9 ⊢ (ℎ = 𝑏 → (ℎ finSupp 0 ↔ 𝑏 finSupp 0))
190		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0})
191	189, 190	elrabrd 3662	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 finSupp 0)
192	188, 191	eqbrtrrd 5136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑖 ∈ 𝐽 ↦ (𝑏‘𝑖)) finSupp 0)
193	77	adantl 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑥 ∈ 𝐵) → (0(.g‘(mulGrp‘𝑅))𝑥) = (1r‘𝑅))
194		fvexd 6894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (1r‘𝑅) ∈ V)
195	192, 193, 181, 185, 194	fsuppssov1 9340	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))) finSupp (1r‘𝑅))
196	35, 37, 170, 173, 187, 195	gsumcl 19981	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))) ∈ 𝐵)
197		fvexd 6894	. . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
198	165, 168, 169, 196, 197	fsuppssov1 9340	. . . 4 ⊢ (𝜑 → (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ ((𝐹‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))))) finSupp (0g‘𝑅))
199		ssidd 3968	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
200	102	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
201	34, 127, 200, 169, 196	ringcld 20338	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ((𝐹‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))) ∈ 𝐵)
202		breq1 5113	. . . . 5 ⊢ (ℎ = (𝑐 ↾ 𝐽) → (ℎ finSupp 0 ↔ (𝑐 ↾ 𝐽) finSupp 0))
203	21, 90	elmapssresd 8861	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 ↾ 𝐽) ∈ (ℕ0 ↑m 𝐽))
204	63	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 0 ∈ ℕ0)
205	27, 204	fsuppres 9349	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 ↾ 𝐽) finSupp 0)
206	202, 203, 205	elrabd 3661	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 ↾ 𝐽) ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0})
207		breq1 5113	. . . . . . 7 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → (ℎ finSupp 0 ↔ (𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0))
208		fveq1 6878	. . . . . . . 8 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → (ℎ‘𝑌) = ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌))
209	208	eqeq1d 2771	. . . . . . 7 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → ((ℎ‘𝑌) = 0 ↔ ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0))
210	207, 209	anbi12d 643	. . . . . 6 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → ((ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0) ↔ ((𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0 ∧ ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0)))
211	8	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
212	14	uneq1i 4126	. . . . . . . . . 10 ⊢ (𝐽 ∪ {𝑌}) = ((𝐼 ∖ {𝑌}) ∪ {𝑌})
213		undifr 4446	. . . . . . . . . . 11 ⊢ ({𝑌} ⊆ 𝐼 ↔ ((𝐼 ∖ {𝑌}) ∪ {𝑌}) = 𝐼)
214	42, 213	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 ∖ {𝑌}) ∪ {𝑌}) = 𝐼)
215	212, 214	eqtrid 2816	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ∪ {𝑌}) = 𝐼)
216	215	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐽 ∪ {𝑌}) = 𝐼)
217	63	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ∈ ℕ0)
218	12, 217	fsnd 6863	. . . . . . . . . 10 ⊢ (𝜑 → {⟨𝑌, 0⟩}:{𝑌}⟶ℕ0)
219	218	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → {⟨𝑌, 0⟩}:{𝑌}⟶ℕ0)
220	14	ineq1i 4177	. . . . . . . . . . 11 ⊢ (𝐽 ∩ {𝑌}) = ((𝐼 ∖ {𝑌}) ∩ {𝑌})
221		disjdifr 4436	. . . . . . . . . . 11 ⊢ ((𝐼 ∖ {𝑌}) ∩ {𝑌}) = ∅
222	220, 221	eqtri 2792	. . . . . . . . . 10 ⊢ (𝐽 ∩ {𝑌}) = ∅
223	222	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐽 ∩ {𝑌}) = ∅)
224	180, 219, 223	fun2d 6740	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}):(𝐽 ∪ {𝑌})⟶ℕ0)
225	216, 224	feq2dd 6689	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}):𝐼⟶ℕ0)
226	176, 211, 225	elmapdd 8834	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}) ∈ (ℕ0 ↑m 𝐼))
227	12, 63	jctir 529	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0))
228	227	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0))
229	180	ffund 6708	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → Fun 𝑏)
230		neldifsnd 4762	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐼 ∖ {𝑌}))
231	14	eleq2i 2861	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ 𝐽 ↔ 𝑌 ∈ (𝐼 ∖ {𝑌}))
232	230, 231	sylnibr 332	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐽)
233	232	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ¬ 𝑌 ∈ 𝐽)
234	180	fdmd 6714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → dom 𝑏 = 𝐽)
235	233, 234	neleqtrrd 2892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ¬ 𝑌 ∈ dom 𝑏)
236		df-nel 3071	. . . . . . . . . 10 ⊢ (𝑌 ∉ dom 𝑏 ↔ ¬ 𝑌 ∈ dom 𝑏)
237	235, 236	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑌 ∉ dom 𝑏)
238	229, 237	jca 520	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (Fun 𝑏 ∧ 𝑌 ∉ dom 𝑏))
239		funsnfsupp 9348	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0) ∧ (Fun 𝑏 ∧ 𝑌 ∉ dom 𝑏)) → ((𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0 ↔ 𝑏 finSupp 0))
240	239	biimpar 482	. . . . . . . 8 ⊢ ((((𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0) ∧ (Fun 𝑏 ∧ 𝑌 ∉ dom 𝑏)) ∧ 𝑏 finSupp 0) → (𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0)
241	228, 238, 191, 240	syl21anc 850	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0)
242	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑌 ∈ 𝐼)
243	63	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
244		fsnunfv 7183	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0 ∧ ¬ 𝑌 ∈ dom 𝑏) → ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0)
245	242, 243, 235, 244	syl3anc 1396	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0)
246	241, 245	jca 520	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ((𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0 ∧ ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0))
247	210, 226, 246	elrabd 3661	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}) ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})
248		simpr 489	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → 𝑏 = (𝑐 ↾ 𝐽))
249	248	uneq1d 4129	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → (𝑏 ∪ {⟨𝑌, 0⟩}) = ((𝑐 ↾ 𝐽) ∪ {⟨𝑌, 0⟩}))
250	14	reseq2i 5973	. . . . . . . . 9 ⊢ (𝑐 ↾ 𝐽) = (𝑐 ↾ (𝐼 ∖ {𝑌}))
251	250	uneq1i 4126	. . . . . . . 8 ⊢ ((𝑐 ↾ 𝐽) ∪ {⟨𝑌, 0⟩}) = ((𝑐 ↾ (𝐼 ∖ {𝑌})) ∪ {⟨𝑌, 0⟩})
252	251	a1i 11	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → ((𝑐 ↾ 𝐽) ∪ {⟨𝑌, 0⟩}) = ((𝑐 ↾ (𝐼 ∖ {𝑌})) ∪ {⟨𝑌, 0⟩}))
253	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → ℕ0 ∈ V)
254	9, 253, 21	elmaprd 32962	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐:𝐼⟶ℕ0)
255	254	ad4ant13 763	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → 𝑐:𝐼⟶ℕ0)
256	255	ffnd 6704	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → 𝑐 Fn 𝐼)
257	242	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → 𝑌 ∈ 𝐼)
258	26	ad4ant13 763	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → (𝑐 finSupp 0 ∧ (𝑐‘𝑌) = 0))
259	258	simprd 500	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → (𝑐‘𝑌) = 0)
260		fresunsn 32907	. . . . . . . 8 ⊢ ((𝑐 Fn 𝐼 ∧ 𝑌 ∈ 𝐼 ∧ (𝑐‘𝑌) = 0) → ((𝑐 ↾ (𝐼 ∖ {𝑌})) ∪ {⟨𝑌, 0⟩}) = 𝑐)
261	256, 257, 259, 260	syl3anc 1396	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → ((𝑐 ↾ (𝐼 ∖ {𝑌})) ∪ {⟨𝑌, 0⟩}) = 𝑐)
262	249, 252, 261	3eqtrrd 2809	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑏 = (𝑐 ↾ 𝐽)) → 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩}))
263		simpr 489	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩}))
264	263	reseq1d 5975	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → (𝑐 ↾ 𝐽) = ((𝑏 ∪ {⟨𝑌, 0⟩}) ↾ 𝐽))
265	180	ad2antrr 738	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → 𝑏:𝐽⟶ℕ0)
266	265	ffnd 6704	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → 𝑏 Fn 𝐽)
267	233	ad2antrr 738	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → ¬ 𝑌 ∈ 𝐽)
268		fsnunres 7184	. . . . . . . 8 ⊢ ((𝑏 Fn 𝐽 ∧ ¬ 𝑌 ∈ 𝐽) → ((𝑏 ∪ {⟨𝑌, 0⟩}) ↾ 𝐽) = 𝑏)
269	266, 267, 268	syl2anc 595	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → ((𝑏 ∪ {⟨𝑌, 0⟩}) ↾ 𝐽) = 𝑏)
270	264, 269	eqtr2d 2805	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})) → 𝑏 = (𝑐 ↾ 𝐽))
271	262, 270	impbida 812	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑏 = (𝑐 ↾ 𝐽) ↔ 𝑐 = (𝑏 ∪ {⟨𝑌, 0⟩})))
272	247, 271	reu6dv 32756	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ∃!𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}𝑏 = (𝑐 ↾ 𝐽))
273	149, 34, 7, 155, 103, 158, 198, 199, 201, 206, 272	gsummptfsf1o 33317	. . 3 ⊢ (𝜑 → (𝑅 Σg (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ ((𝐹‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))) = (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)} ↦ ((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))))
274	101, 148, 273	3eqtr4d 2814	. 2 ⊢ (𝜑 → (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))) = (𝑅 Σg (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ ((𝐹‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))))
275		evlextv.q	. . . 4 ⊢ 𝑄 = (𝐼 eval 𝑅)
276	275, 34	evlval 22216	. . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
277		eqid 2769	. . 3 ⊢ (𝐼 mPoly (𝑅 ↾s 𝐵)) = (𝐼 mPoly (𝑅 ↾s 𝐵))
278		eqid 2769	. . 3 ⊢ (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵)))
279		eqid 2769	. . 3 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
280	34	subrgid 20654	. . . 4 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
281	102, 280	syl 18	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
282	34	ressid 17300	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅)
283	10, 282	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾s 𝐵) = 𝑅)
284	283	oveq2d 7424	. . . . 5 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾s 𝐵)) = (𝐼 mPoly 𝑅))
285	284	fveq2d 6883	. . . 4 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐼 mPoly 𝑅)))
286	138, 285	eleqtrrd 2872	. . 3 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹) ∈ (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵))))
287	34	fvexi 6893	. . . . 5 ⊢ 𝐵 ∈ V
288	287	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
289	288, 8, 53	elmapdd 8834	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐼))
290	276, 277, 278, 279, 136, 34, 33, 58, 127, 8, 10, 281, 286, 289	evlsvvval 22209	. 2 ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = (𝑅 Σg (𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ↦ ((((𝐸‘𝑌)‘𝐹)‘𝑐)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐼 ↦ ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖))))))))
291		evlextv.o	. . . 4 ⊢ 𝑂 = (𝐽 eval 𝑅)
292	291, 34	evlval 22216	. . 3 ⊢ 𝑂 = ((𝐽 evalSub 𝑅)‘𝐵)
293		eqid 2769	. . 3 ⊢ (𝐽 mPoly (𝑅 ↾s 𝐵)) = (𝐽 mPoly (𝑅 ↾s 𝐵))
294		eqid 2769	. . 3 ⊢ (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵)))
295	16, 15	eleqtrdi 2879	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐽 mPoly 𝑅)))
296	283	oveq2d 7424	. . . . 5 ⊢ (𝜑 → (𝐽 mPoly (𝑅 ↾s 𝐵)) = (𝐽 mPoly 𝑅))
297	296	fveq2d 6883	. . . 4 ⊢ (𝜑 → (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐽 mPoly 𝑅)))
298	295, 297	eleqtrrd 2872	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵))))
299	289, 171	elmapssresd 8861	. . 3 ⊢ (𝜑 → (𝐴 ↾ 𝐽) ∈ (𝐵 ↑m 𝐽))
300	292, 293, 294, 279, 161, 34, 33, 58, 127, 172, 10, 281, 298, 299	evlsvvval 22209	. 2 ⊢ (𝜑 → ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)) = (𝑅 Σg (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ ((𝐹‘𝑏)(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖))))))))
301	274, 290, 300	3eqtr4d 2814	1 ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∉ wnel 3070  {crab 3423  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  ifcif 4489  {csn 4591  ⟨cop 4597   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5659   ↾ cres 5661  Fun wfun 6528   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408   ↑_m cmap 8820   finSupp cfsupp 9317  0cc0 11096  ℕ₀cn0 12500  Basecbs 17265   ↾_s cress 17286  ._rcmulr 17307  0_gc0g 17488   Σ_g cgsu 17489  Mndcmnd 18788  ._gcmg 19129  CMndccmn 19846  mulGrpcmgp 20212  1_rcur 20259  Ringcrg 20311  CRingccrg 20312  SubRingcsubrg 20650   mPoly cmpl 22021   eval cevl 22189  extendVarscextv 33860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-ofr 7673  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-pm 8823  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-card 9921  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-fzo 13679  df-seq 14034  df-hash 14363  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-hom 17330  df-cco 17331  df-0g 17490  df-gsum 17491  df-prds 17496  df-pws 17498  df-mre 17634  df-mrc 17635  df-acs 17637  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-mhm 18837  df-submnd 18838  df-grp 18999  df-minusg 19000  df-sbg 19001  df-mulg 19130  df-subg 19185  df-ghm 19280  df-cntz 19383  df-cmn 19848  df-abl 19849  df-mgp 20213  df-rng 20227  df-ur 20260  df-srg 20265  df-ring 20313  df-cring 20314  df-rhm 20550  df-subrng 20627  df-subrg 20651  df-lmod 20957  df-lss 21027  df-lsp 21067  df-assa 21968  df-asp 21969  df-ascl 21970  df-psr 22024  df-mvr 22025  df-mpl 22026  df-evls 22190  df-evl 22191  df-extv 33861

This theorem is referenced by:  esplyindfv  33907