Theorem evlextv 33686

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 6841	. . . . . . . . . 10 ⊢ (𝐸‘𝑌) = ((𝐼extendVars𝑅)‘𝑌)
3	2	fveq1i 6841	. . . . . . . . 9 ⊢ ((𝐸‘𝑌)‘𝐹) = (((𝐼extendVars𝑅)‘𝑌)‘𝐹)
4	3	fveq1i 6841	. . . . . . . 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 2736	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}
7		eqid 2736	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑅)
8		evlextv.i	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
9	8	adantr 480	. . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝐹 ∈ 𝑀)
18		breq1 5088	. . . . . . . . 9 ⊢ (ℎ = 𝑐 → (ℎ finSupp 0 ↔ 𝑐 finSupp 0))
19		ssrab2 4020	. . . . . . . . . . 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 3921	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐 ∈ (ℕ0 ↑m 𝐼))
22		fveq1 6839	. . . . . . . . . . . . 13 ⊢ (ℎ = 𝑐 → (ℎ‘𝑌) = (𝑐‘𝑌))
23	22	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (ℎ = 𝑐 → ((ℎ‘𝑌) = 0 ↔ (𝑐‘𝑌) = 0))
24	18, 23	anbi12d 633	. . . . . . . . . . 11 ⊢ (ℎ = 𝑐 → ((ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0) ↔ (𝑐 finSupp 0 ∧ (𝑐‘𝑌) = 0)))
25		simpr 484	. . . . . . . . . . 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 32568	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 finSupp 0 ∧ (𝑐‘𝑌) = 0))
27	26	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐 finSupp 0)
28	18, 21, 27	elrabd 3636	. . . . . . . 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 33678	. . . . . . 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 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐‘𝑌) = 0)
31	30	iftrued 4474	. . . . . . 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 2775	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (((𝐸‘𝑌)‘𝐹)‘𝑐) = (𝐹‘(𝑐 ↾ 𝐽)))
33		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
34		evlextv.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
35	33, 34	mgpbas 20126	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
36		eqid 2736	. . . . . . . . 9 ⊢ (1r‘𝑅) = (1r‘𝑅)
37	33, 36	ringidval 20164	. . . . . . . 8 ⊢ (1r‘𝑅) = (0g‘(mulGrp‘𝑅))
38	33	crngmgp 20222	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → (mulGrp‘𝑅) ∈ CMnd)
39	11, 38	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (mulGrp‘𝑅) ∈ CMnd)
40		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 ∈ (𝐼 ∖ 𝐽))
41	14	difeq2i 4063	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∖ 𝐽) = (𝐼 ∖ (𝐼 ∖ {𝑌}))
42	12	snssd 4730	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {𝑌} ⊆ 𝐼)
43		dfss4 4209	. . . . . . . . . . . . . . . . 17 ⊢ ({𝑌} ⊆ 𝐼 ↔ (𝐼 ∖ (𝐼 ∖ {𝑌})) = {𝑌})
44	42, 43	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐼 ∖ (𝐼 ∖ {𝑌})) = {𝑌})
45	41, 44	eqtrid 2783	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐼 ∖ 𝐽) = {𝑌})
46	45	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝐼 ∖ 𝐽) = {𝑌})
47	40, 46	eleqtrd 2838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 ∈ {𝑌})
48	47	elsnd 4585	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 = 𝑌)
49	48	fveq2d 6844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑖) = (𝑐‘𝑌))
50	30	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑌) = 0)
51	49, 50	eqtrd 2771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝑐‘𝑖) = 0)
52	51	oveq1d 7382	. . . . . . . . 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 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝐴:𝐼⟶𝐵)
55		difssd 4077	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝐼 ∖ 𝐽) ⊆ 𝐼)
56	55	sselda 3921	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → 𝑖 ∈ 𝐼)
57	54, 56	ffvelcdmd 7037	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ (𝐼 ∖ 𝐽)) → (𝐴‘𝑖) ∈ 𝐵)
58		eqid 2736	. . . . . . . . . . 11 ⊢ (.g‘(mulGrp‘𝑅)) = (.g‘(mulGrp‘𝑅))
59	35, 37, 58	mulg0 19050	. . . . . . . . . 10 ⊢ ((𝐴‘𝑖) ∈ 𝐵 → (0(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) = (1r‘𝑅))
60	57, 59	syl 17	. . . . . . . . 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 2771	. . . . . . . 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 6855	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (1r‘𝑅) ∈ V)
63		0nn0 12452	. . . . . . . . . . 11 ⊢ 0 ∈ ℕ0
64	63	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 0 ∈ ℕ0)
65	8	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
66		ssidd 3945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐼 ⊆ 𝐼)
67	53	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝐴:𝐼⟶𝐵)
68	67	ffvelcdmda 7036	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) ∈ 𝐵)
69		nn0ex 12443	. . . . . . . . . . . 12 ⊢ ℕ0 ∈ V
70	69	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → ℕ0 ∈ V)
71		ssrab2 4020	. . . . . . . . . . . . 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 3921	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐 ∈ (ℕ0 ↑m 𝐼))
74	65, 70, 73	elmaprd 32753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐:𝐼⟶ℕ0)
75		simpr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0})
76	18, 75	elrabrd 32568	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑐 finSupp 0)
77	35, 37, 58	mulg0 19050	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 → (0(.g‘(mulGrp‘𝑅))𝑥) = (1r‘𝑅))
78	77	adantl 481	. . . . . . . . . 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 32756	. . . . . . . . 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 592	. . . . . . . 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 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝑅) ∈ CMnd)
82	81	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ CMnd)
83	82	cmnmndd 19779	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ Mnd)
84	83	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑅) ∈ Mnd)
85	74	ffvelcdmda 7036	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → (𝑐‘𝑖) ∈ ℕ0)
86	35, 58, 84, 85, 68	mulgnn0cld 19071	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐼) → ((𝑐‘𝑖)(.g‘(mulGrp‘𝑅))(𝐴‘𝑖)) ∈ 𝐵)
87	28, 86	syldanl 603	. . . . . . . 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 4076	. . . . . . . . . 10 ⊢ (𝐼 ∖ {𝑌}) ⊆ 𝐼
89	14, 88	eqsstri 3968	. . . . . . . . 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 33115	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → 𝑖 ∈ 𝐽)
93	92	fvresd 6860	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → ((𝑐 ↾ 𝐽)‘𝑖) = (𝑐‘𝑖))
94	92	fvresd 6860	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) ∧ 𝑖 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑖) = (𝐴‘𝑖))
95	93, 94	oveq12d 7385	. . . . . . . . 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 5178	. . . . . . . 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 7383	. . . . . . 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 2774	. . . . . 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 7385	. . . . 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 5178	. . . 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 7383	. . 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 20227	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring)
103	102	ringcmnd 20265	. . . 4 ⊢ (𝜑 → 𝑅 ∈ CMnd)
104		ovex 7400	. . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V
105	104	rabex 5280	. . . . 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 480	. . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ ({ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0} ∖ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)})) → 𝐹 ∈ 𝑀)
112		difssd 4077	. . . . . . . . 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 3921	. . . . . . . 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 33678	. . . . . . 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 480	. . . . . . . . . . 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 3919	. . . . . . . . . 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 32568	. . . . . . . . . . 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 484	. . . . . . . . . . 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 511	. . . . . . . . . 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 3636	. . . . . . . . 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 769	. . . . . . . . . 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 3902	. . . . . . . . 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 817	. . . . . . . 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 4477	. . . . . . 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 2775	. . . . . 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 7382	. . . . 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 2736	. . . . . 6 ⊢ (.r‘𝑅) = (.r‘𝑅)
128	102	adantr 480	. . . . . 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 7067	. . . . . . . 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 19890	. . . . . . 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 592	. . . . . 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 20276	. . . . 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 2771	. . . 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 2736	. . . . . 6 ⊢ (𝐼 mPoly 𝑅) = (𝐼 mPoly 𝑅)
135		eqid 2736	. . . . . 6 ⊢ (Base‘(𝐼 mPoly 𝑅)) = (Base‘(𝐼 mPoly 𝑅))
136	6	psrbasfsupp 33672	. . . . . 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 33680	. . . . . . 7 ⊢ (𝜑 → (((𝐼extendVars𝑅)‘𝑌)‘𝐹) ∈ (Base‘(𝐼 mPoly 𝑅)))
138	3, 137	eqeltrid 2840	. . . . . 6 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹) ∈ (Base‘(𝐼 mPoly 𝑅)))
139	134, 34, 135, 136, 138	mplelf 21976	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹):{ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}⟶𝐵)
140	134, 135, 7, 138	mplelsfi 21973	. . . . 5 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹) finSupp (0g‘𝑅))
141	34, 102, 106, 130, 139, 140	rmfsupp2 33299	. . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
143	139	ffvelcdmda 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ ℎ finSupp 0}) → (((𝐸‘𝑌)‘𝐹)‘𝑐) ∈ 𝐵)
144	34, 127, 142, 143, 130	ringcld 20241	. . . 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 482	. . . . . 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 4015	. . . 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 33115	. . 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 2898	. . . 4 ⊢ Ⅎ𝑏((𝐹‘(𝑐 ↾ 𝐽))(.r‘𝑅)((mulGrp‘𝑅) Σg (𝑖 ∈ 𝐽 ↦ (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))))
150		fveq2 6840	. . . . 5 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → (𝐹‘𝑏) = (𝐹‘(𝑐 ↾ 𝐽)))
151		fveq1 6839	. . . . . . . 8 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → (𝑏‘𝑖) = ((𝑐 ↾ 𝐽)‘𝑖))
152	151	oveq1d 7382	. . . . . . 7 ⊢ (𝑏 = (𝑐 ↾ 𝐽) → ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)) = (((𝑐 ↾ 𝐽)‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)))
153	152	mpteq2dv 5179	. . . . . 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 7383	. . . . 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 7385	. . . 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 7400	. . . . . 6 ⊢ (ℕ0 ↑m 𝐽) ∈ V
157	156	rabex 5280	. . . . 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 2736	. . . . . . . 8 ⊢ (𝐽 mPoly 𝑅) = (𝐽 mPoly 𝑅)
160		eqid 2736	. . . . . . . . 9 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}
161	160	psrbasfsupp 33672	. . . . . . . 8 ⊢ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} = {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ (◡ℎ “ ℕ) ∈ Fin}
162	159, 34, 15, 161, 16	mplelf 21976	. . . . . . 7 ⊢ (𝜑 → 𝐹:{ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}⟶𝐵)
163	162	feqmptd 6908	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ (𝐹‘𝑏)))
164	159, 15, 7, 16	mplelsfi 21973	. . . . . 6 ⊢ (𝜑 → 𝐹 finSupp (0g‘𝑅))
165	163, 164	eqbrtrrd 5109	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0} ↦ (𝐹‘𝑏)) finSupp (0g‘𝑅))
166	102	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ Ring)
167		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
168	34, 127, 7, 166, 167	ringlzd 20276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((0g‘𝑅)(.r‘𝑅)𝑥) = (0g‘𝑅))
169	162	ffvelcdmda 7036	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐹‘𝑏) ∈ 𝐵)
170	81	adantr 480	. . . . . 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 5265	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ V)
173	172	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝐽 ∈ V)
174	170	cmnmndd 19779	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (mulGrp‘𝑅) ∈ Mnd)
175	174	adantr 480	. . . . . . . 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 4020	. . . . . . . . . . . 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 3921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 ∈ (ℕ0 ↑m 𝐽))
180	173, 176, 179	elmaprd 32753	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏:𝐽⟶ℕ0)
181	180	ffvelcdmda 7036	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → (𝑏‘𝑖) ∈ ℕ0)
182	53	adantr 480	. . . . . . . . . 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 6707	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐴 ↾ 𝐽):𝐽⟶𝐵)
185	184	ffvelcdmda 7036	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → ((𝐴 ↾ 𝐽)‘𝑖) ∈ 𝐵)
186	35, 58, 175, 181, 185	mulgnn0cld 19071	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑖 ∈ 𝐽) → ((𝑏‘𝑖)(.g‘(mulGrp‘𝑅))((𝐴 ↾ 𝐽)‘𝑖)) ∈ 𝐵)
187	186	fmpttd 7067	. . . . . 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 6908	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 = (𝑖 ∈ 𝐽 ↦ (𝑏‘𝑖)))
189		breq1 5088	. . . . . . . . 9 ⊢ (ℎ = 𝑏 → (ℎ finSupp 0 ↔ 𝑏 finSupp 0))
190		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0})
191	189, 190	elrabrd 32568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑏 finSupp 0)
192	188, 191	eqbrtrrd 5109	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑖 ∈ 𝐽 ↦ (𝑏‘𝑖)) finSupp 0)
193	77	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) ∧ 𝑥 ∈ 𝐵) → (0(.g‘(mulGrp‘𝑅))𝑥) = (1r‘𝑅))
194		fvexd 6855	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (1r‘𝑅) ∈ V)
195	192, 193, 181, 185, 194	fsuppssov1 9297	. . . . . 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 19890	. . . . 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 6855	. . . . 5 ⊢ (𝜑 → (0g‘𝑅) ∈ V)
198	165, 168, 169, 196, 197	fsuppssov1 9297	. . . 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 3945	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐵)
200	102	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑅 ∈ Ring)
201	34, 127, 200, 169, 196	ringcld 20241	. . . 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 5088	. . . . 5 ⊢ (ℎ = (𝑐 ↾ 𝐽) → (ℎ finSupp 0 ↔ (𝑐 ↾ 𝐽) finSupp 0))
203	21, 90	elmapssresd 8815	. . . . 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 9306	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 ↾ 𝐽) finSupp 0)
206	202, 203, 205	elrabd 3636	. . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → (𝑐 ↾ 𝐽) ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0})
207		breq1 5088	. . . . . . 7 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → (ℎ finSupp 0 ↔ (𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0))
208		fveq1 6839	. . . . . . . 8 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → (ℎ‘𝑌) = ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌))
209	208	eqeq1d 2738	. . . . . . 7 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → ((ℎ‘𝑌) = 0 ↔ ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0))
210	207, 209	anbi12d 633	. . . . . 6 ⊢ (ℎ = (𝑏 ∪ {⟨𝑌, 0⟩}) → ((ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0) ↔ ((𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0 ∧ ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0)))
211	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝐼 ∈ 𝑉)
212	14	uneq1i 4104	. . . . . . . . . 10 ⊢ (𝐽 ∪ {𝑌}) = ((𝐼 ∖ {𝑌}) ∪ {𝑌})
213		undifr 4423	. . . . . . . . . . 11 ⊢ ({𝑌} ⊆ 𝐼 ↔ ((𝐼 ∖ {𝑌}) ∪ {𝑌}) = 𝐼)
214	42, 213	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐼 ∖ {𝑌}) ∪ {𝑌}) = 𝐼)
215	212, 214	eqtrid 2783	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ∪ {𝑌}) = 𝐼)
216	215	adantr 480	. . . . . . . 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 6824	. . . . . . . . . 10 ⊢ (𝜑 → {⟨𝑌, 0⟩}:{𝑌}⟶ℕ0)
219	218	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → {⟨𝑌, 0⟩}:{𝑌}⟶ℕ0)
220	14	ineq1i 4156	. . . . . . . . . . 11 ⊢ (𝐽 ∩ {𝑌}) = ((𝐼 ∖ {𝑌}) ∩ {𝑌})
221		disjdifr 4413	. . . . . . . . . . 11 ⊢ ((𝐼 ∖ {𝑌}) ∩ {𝑌}) = ∅
222	220, 221	eqtri 2759	. . . . . . . . . 10 ⊢ (𝐽 ∩ {𝑌}) = ∅
223	222	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝐽 ∩ {𝑌}) = ∅)
224	180, 219, 223	fun2d 6704	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}):(𝐽 ∪ {𝑌})⟶ℕ0)
225	216, 224	feq2dd 6654	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}):𝐼⟶ℕ0)
226	176, 211, 225	elmapdd 8788	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}) ∈ (ℕ0 ↑m 𝐼))
227	12, 63	jctir 520	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0))
228	227	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0))
229	180	ffund 6672	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → Fun 𝑏)
230		neldifsnd 4738	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝐼 ∖ {𝑌}))
231	14	eleq2i 2828	. . . . . . . . . . . . 13 ⊢ (𝑌 ∈ 𝐽 ↔ 𝑌 ∈ (𝐼 ∖ {𝑌}))
232	230, 231	sylnibr 329	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑌 ∈ 𝐽)
233	232	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ¬ 𝑌 ∈ 𝐽)
234	180	fdmd 6678	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → dom 𝑏 = 𝐽)
235	233, 234	neleqtrrd 2859	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ¬ 𝑌 ∈ dom 𝑏)
236		df-nel 3037	. . . . . . . . . 10 ⊢ (𝑌 ∉ dom 𝑏 ↔ ¬ 𝑌 ∈ dom 𝑏)
237	235, 236	sylibr 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → 𝑌 ∉ dom 𝑏)
238	229, 237	jca 511	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (Fun 𝑏 ∧ 𝑌 ∉ dom 𝑏))
239		funsnfsupp 9305	. . . . . . . . 9 ⊢ (((𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0) ∧ (Fun 𝑏 ∧ 𝑌 ∉ dom 𝑏)) → ((𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0 ↔ 𝑏 finSupp 0))
240	239	biimpar 477	. . . . . . . 8 ⊢ ((((𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0) ∧ (Fun 𝑏 ∧ 𝑌 ∉ dom 𝑏)) ∧ 𝑏 finSupp 0) → (𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0)
241	228, 238, 191, 240	syl21anc 838	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → (𝑏 ∪ {⟨𝑌, 0⟩}) finSupp 0)
242	12	adantr 480	. . . . . . . 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 7142	. . . . . . . 8 ⊢ ((𝑌 ∈ 𝐼 ∧ 0 ∈ ℕ0 ∧ ¬ 𝑌 ∈ dom 𝑏) → ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0)
245	242, 243, 235, 244	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ0 ↑m 𝐽) ∣ ℎ finSupp 0}) → ((𝑏 ∪ {⟨𝑌, 0⟩})‘𝑌) = 0)
246	241, 245	jca 511	. . . . . 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 3636	. . . . 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 484	. . . . . . . 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 4107	. . . . . . 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 5941	. . . . . . . . 9 ⊢ (𝑐 ↾ 𝐽) = (𝑐 ↾ (𝐼 ∖ {𝑌}))
251	250	uneq1i 4104	. . . . . . . 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 32753	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑐 ∈ {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (ℎ finSupp 0 ∧ (ℎ‘𝑌) = 0)}) → 𝑐:𝐼⟶ℕ0)
255	254	ad4ant13 752	. . . . . . . . 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 6669	. . . . . . . 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 727	. . . . . . . 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 752	. . . . . . . . 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 495	. . . . . . . 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 32698	. . . . . . . 8 ⊢ ((𝑐 Fn 𝐼 ∧ 𝑌 ∈ 𝐼 ∧ (𝑐‘𝑌) = 0) → ((𝑐 ↾ (𝐼 ∖ {𝑌})) ∪ {⟨𝑌, 0⟩}) = 𝑐)
261	256, 257, 259, 260	syl3anc 1374	. . . . . . 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 2776	. . . . . 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 484	. . . . . . . 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 5943	. . . . . . 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 727	. . . . . . . . 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 6669	. . . . . . . 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 727	. . . . . . . 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 7143	. . . . . . . 8 ⊢ ((𝑏 Fn 𝐽 ∧ ¬ 𝑌 ∈ 𝐽) → ((𝑏 ∪ {⟨𝑌, 0⟩}) ↾ 𝐽) = 𝑏)
269	266, 267, 268	syl2anc 585	. . . . . . 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 2772	. . . . . 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 801	. . . . 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 32542	. . . 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 33121	. . 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 2781	. 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 22078	. . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
277		eqid 2736	. . 3 ⊢ (𝐼 mPoly (𝑅 ↾s 𝐵)) = (𝐼 mPoly (𝑅 ↾s 𝐵))
278		eqid 2736	. . 3 ⊢ (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵)))
279		eqid 2736	. . 3 ⊢ (𝑅 ↾s 𝐵) = (𝑅 ↾s 𝐵)
280	34	subrgid 20550	. . . 4 ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
281	102, 280	syl 17	. . 3 ⊢ (𝜑 → 𝐵 ∈ (SubRing‘𝑅))
282	34	ressid 17214	. . . . . . 7 ⊢ (𝑅 ∈ CRing → (𝑅 ↾s 𝐵) = 𝑅)
283	10, 282	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑅 ↾s 𝐵) = 𝑅)
284	283	oveq2d 7383	. . . . 5 ⊢ (𝜑 → (𝐼 mPoly (𝑅 ↾s 𝐵)) = (𝐼 mPoly 𝑅))
285	284	fveq2d 6844	. . . 4 ⊢ (𝜑 → (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐼 mPoly 𝑅)))
286	138, 285	eleqtrrd 2839	. . 3 ⊢ (𝜑 → ((𝐸‘𝑌)‘𝐹) ∈ (Base‘(𝐼 mPoly (𝑅 ↾s 𝐵))))
287	34	fvexi 6854	. . . . 5 ⊢ 𝐵 ∈ V
288	287	a1i 11	. . . 4 ⊢ (𝜑 → 𝐵 ∈ V)
289	288, 8, 53	elmapdd 8788	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ↑m 𝐼))
290	276, 277, 278, 279, 136, 34, 33, 58, 127, 8, 10, 281, 286, 289	evlsvvval 22071	. 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 22078	. . 3 ⊢ 𝑂 = ((𝐽 evalSub 𝑅)‘𝐵)
293		eqid 2736	. . 3 ⊢ (𝐽 mPoly (𝑅 ↾s 𝐵)) = (𝐽 mPoly (𝑅 ↾s 𝐵))
294		eqid 2736	. . 3 ⊢ (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵)))
295	16, 15	eleqtrdi 2846	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐽 mPoly 𝑅)))
296	283	oveq2d 7383	. . . . 5 ⊢ (𝜑 → (𝐽 mPoly (𝑅 ↾s 𝐵)) = (𝐽 mPoly 𝑅))
297	296	fveq2d 6844	. . . 4 ⊢ (𝜑 → (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵))) = (Base‘(𝐽 mPoly 𝑅)))
298	295, 297	eleqtrrd 2839	. . 3 ⊢ (𝜑 → 𝐹 ∈ (Base‘(𝐽 mPoly (𝑅 ↾s 𝐵))))
299	289, 171	elmapssresd 8815	. . 3 ⊢ (𝜑 → (𝐴 ↾ 𝐽) ∈ (𝐵 ↑m 𝐽))
300	292, 293, 294, 279, 161, 34, 33, 58, 127, 172, 10, 281, 298, 299	evlsvvval 22071	. 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 2781	1 ⊢ (𝜑 → ((𝑄‘((𝐸‘𝑌)‘𝐹))‘𝐴) = ((𝑂‘𝐹)‘(𝐴 ↾ 𝐽)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∉ wnel 3036  {crab 3389  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ifcif 4466  {csn 4567  ⟨cop 4573   class class class wbr 5085   ↦ cmpt 5166  dom cdm 5631   ↾ cres 5633  Fun wfun 6492   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   finSupp cfsupp 9274  0cc0 11038  ℕ₀cn0 12437  Basecbs 17179   ↾_s cress 17200  ._rcmulr 17221  0_gc0g 17402   Σ_g cgsu 17403  Mndcmnd 18702  ._gcmg 19043  CMndccmn 19755  mulGrpcmgp 20121  1_rcur 20162  Ringcrg 20214  CRingccrg 20215  SubRingcsubrg 20546   mPoly cmpl 21886   eval cevl 22051  extendVarscextv 33673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-ofr 7632  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-sup 9355  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-fzo 13609  df-seq 13964  df-hash 14293  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-hom 17244  df-cco 17245  df-0g 17404  df-gsum 17405  df-prds 17410  df-pws 17412  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-grp 18912  df-minusg 18913  df-sbg 18914  df-mulg 19044  df-subg 19099  df-ghm 19188  df-cntz 19292  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-srg 20168  df-ring 20216  df-cring 20217  df-rhm 20452  df-subrng 20523  df-subrg 20547  df-lmod 20857  df-lss 20927  df-lsp 20967  df-assa 21833  df-asp 21834  df-ascl 21835  df-psr 21889  df-mvr 21890  df-mpl 21891  df-evls 22052  df-evl 22053  df-extv 33674

This theorem is referenced by:  esplyindfv  33720