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Theorem coe1fzgsumdlem 21816

Description: Lemma for coe1fzgsumd 21817 (induction step). (Contributed by AV, 8-Oct-2019.)

**Hypotheses**
Ref	Expression
coe1fzgsumd.p	⊢ 𝑃 = (Poly₁‘𝑅)
coe1fzgsumd.b	⊢ 𝐵 = (Base‘𝑃)
coe1fzgsumd.r	⊢ (𝜑 → 𝑅 ∈ Ring)
coe1fzgsumd.k	⊢ (𝜑 → 𝐾 ∈ ℕ₀)

**Assertion**
Ref	Expression
coe1fzgsumdlem	⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐾 𝑥,𝑚 𝑥,𝑎 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑎) 𝐵(𝑚,𝑎) 𝑃(𝑥,𝑚,𝑎) 𝑅(𝑥,𝑚,𝑎) 𝐾(𝑚,𝑎) 𝑀(𝑥,𝑚,𝑎)

Proof of Theorem coe1fzgsumdlem Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralunb 4190	. . 3 ⊢ (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 ↔ (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵))
2		nfcv 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑦𝑀
3		nfcsb1v 3917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑀
4		csbeq1a 3906	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → 𝑀 = ⦋𝑦 / 𝑥⦌𝑀)
5	2, 3, 4	cbvmpt 5258	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀) = (𝑦 ∈ (𝑚 ∪ {𝑎}) ↦ ⦋𝑦 / 𝑥⦌𝑀)
6	5	oveq2i 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)) = (𝑃 Σ_g (𝑦 ∈ (𝑚 ∪ {𝑎}) ↦ ⦋𝑦 / 𝑥⦌𝑀))
7		coe1fzgsumd.b	. . . . . . . . . . . . . . . 16 ⊢ 𝐵 = (Base‘𝑃)
8		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (+_g‘𝑃) = (+_g‘𝑃)
9		coe1fzgsumd.r	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ∈ Ring)
10		coe1fzgsumd.p	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑃 = (Poly₁‘𝑅)
11	10	ply1ring 21761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
12	9, 11	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ Ring)
13		ringcmn 20092	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
14	12, 13	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ CMnd)
15	14	3ad2ant3 1135	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → 𝑃 ∈ CMnd)
16	15	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → 𝑃 ∈ CMnd)
17		simpll1 1212	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → 𝑚 ∈ Fin)
18		rspcsbela 4434	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑚 ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → ⦋𝑦 / 𝑥⦌𝑀 ∈ 𝐵)
19	18	expcom 414	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → (𝑦 ∈ 𝑚 → ⦋𝑦 / 𝑥⦌𝑀 ∈ 𝐵))
20	19	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → (𝑦 ∈ 𝑚 → ⦋𝑦 / 𝑥⦌𝑀 ∈ 𝐵))
21	20	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑦 ∈ 𝑚 → ⦋𝑦 / 𝑥⦌𝑀 ∈ 𝐵))
22	21	imp 407	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ 𝑦 ∈ 𝑚) → ⦋𝑦 / 𝑥⦌𝑀 ∈ 𝐵)
23		vex 3478	. . . . . . . . . . . . . . . . 17 ⊢ 𝑎 ∈ V
24	23	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → 𝑎 ∈ V)
25		simpll2 1213	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ¬ 𝑎 ∈ 𝑚)
26		vsnid 4664	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑎 ∈ {𝑎}
27		rspcsbela 4434	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ {𝑎} ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ⦋𝑎 / 𝑥⦌𝑀 ∈ 𝐵)
28	26, 27	mpan 688	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵 → ⦋𝑎 / 𝑥⦌𝑀 ∈ 𝐵)
29	28	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ⦋𝑎 / 𝑥⦌𝑀 ∈ 𝐵)
30		csbeq1 3895	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑎 → ⦋𝑦 / 𝑥⦌𝑀 = ⦋𝑎 / 𝑥⦌𝑀)
31	7, 8, 16, 17, 22, 24, 25, 29, 30	gsumunsn 19822	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑃 Σ_g (𝑦 ∈ (𝑚 ∪ {𝑎}) ↦ ⦋𝑦 / 𝑥⦌𝑀)) = ((𝑃 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀))
32	6, 31	eqtrid 2784	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)) = ((𝑃 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀))
33	2, 3, 4	cbvmpt 5258	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ 𝑚 ↦ 𝑀) = (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌𝑀)
34	33	eqcomi 2741	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌𝑀) = (𝑥 ∈ 𝑚 ↦ 𝑀)
35	34	oveq2i 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑃 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌𝑀)) = (𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))
36	35	oveq1i 7415	. . . . . . . . . . . . . 14 ⊢ ((𝑃 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀) = ((𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀)
37	32, 36	eqtrdi 2788	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)) = ((𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀))
38	37	fveq2d 6892	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀))) = (coe₁‘((𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀)))
39	38	fveq1d 6890	. . . . . . . . . . 11 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = ((coe₁‘((𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀))‘𝐾))
40	9	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → 𝑅 ∈ Ring)
41	40	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
42		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵)
43	7, 16, 17, 42	gsummptcl 19829	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)) ∈ 𝐵)
44		coe1fzgsumd.k	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
45	44	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → 𝐾 ∈ ℕ₀)
46	45	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → 𝐾 ∈ ℕ₀)
47		eqid 2732	. . . . . . . . . . . . 13 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
48	10, 7, 8, 47	coe1addfv 21778	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ (𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)) ∈ 𝐵 ∧ ⦋𝑎 / 𝑥⦌𝑀 ∈ 𝐵) ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘((𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀))‘𝐾) = (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾)(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
49	41, 43, 29, 46, 48	syl31anc 1373	. . . . . . . . . . 11 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ((coe₁‘((𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀))(+_g‘𝑃)⦋𝑎 / 𝑥⦌𝑀))‘𝐾) = (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾)(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
50	39, 49	eqtrd 2772	. . . . . . . . . 10 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾)(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
51		oveq1 7412	. . . . . . . . . 10 ⊢ (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))) → (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾)(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)) = ((𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
52	50, 51	sylan9eq 2792	. . . . . . . . 9 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))) → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = ((𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
53		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦((coe₁‘𝑀)‘𝐾)
54		nfcsb1v 3917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾)
55		csbeq1a 3906	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((coe₁‘𝑀)‘𝐾) = ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾))
56	53, 54, 55	cbvmpt 5258	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾)) = (𝑦 ∈ (𝑚 ∪ {𝑎}) ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾))
57	56	oveq2i 7416	. . . . . . . . . . . 12 ⊢ (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))) = (𝑅 Σ_g (𝑦 ∈ (𝑚 ∪ {𝑎}) ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾)))
58		eqid 2732	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
59		ringcmn 20092	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
60	9, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ CMnd)
61	60	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → 𝑅 ∈ CMnd)
62	61	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → 𝑅 ∈ CMnd)
63		csbfv12 6936	. . . . . . . . . . . . . . 15 ⊢ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾) = (⦋𝑦 / 𝑥⦌(coe₁‘𝑀)‘⦋𝑦 / 𝑥⦌𝐾)
64		csbfv2g 6937	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌(coe₁‘𝑀) = (coe₁‘⦋𝑦 / 𝑥⦌𝑀))
65	64	elv 3480	. . . . . . . . . . . . . . . 16 ⊢ ⦋𝑦 / 𝑥⦌(coe₁‘𝑀) = (coe₁‘⦋𝑦 / 𝑥⦌𝑀)
66		csbconstg 3911	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌𝐾 = 𝐾)
67	66	elv 3480	. . . . . . . . . . . . . . . 16 ⊢ ⦋𝑦 / 𝑥⦌𝐾 = 𝐾
68	65, 67	fveq12i 6894	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑦 / 𝑥⦌(coe₁‘𝑀)‘⦋𝑦 / 𝑥⦌𝐾) = ((coe₁‘⦋𝑦 / 𝑥⦌𝑀)‘𝐾)
69	63, 68	eqtri 2760	. . . . . . . . . . . . . 14 ⊢ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾) = ((coe₁‘⦋𝑦 / 𝑥⦌𝑀)‘𝐾)
70		eqid 2732	. . . . . . . . . . . . . . . . 17 ⊢ (coe₁‘⦋𝑦 / 𝑥⦌𝑀) = (coe₁‘⦋𝑦 / 𝑥⦌𝑀)
71	70, 7, 10, 58	coe1f 21726	. . . . . . . . . . . . . . . 16 ⊢ (⦋𝑦 / 𝑥⦌𝑀 ∈ 𝐵 → (coe₁‘⦋𝑦 / 𝑥⦌𝑀):ℕ₀⟶(Base‘𝑅))
72	22, 71	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ 𝑦 ∈ 𝑚) → (coe₁‘⦋𝑦 / 𝑥⦌𝑀):ℕ₀⟶(Base‘𝑅))
73	45	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → 𝐾 ∈ ℕ₀)
74	73	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ 𝑦 ∈ 𝑚) → 𝐾 ∈ ℕ₀)
75	72, 74	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ 𝑦 ∈ 𝑚) → ((coe₁‘⦋𝑦 / 𝑥⦌𝑀)‘𝐾) ∈ (Base‘𝑅))
76	69, 75	eqeltrid 2837	. . . . . . . . . . . . 13 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ 𝑦 ∈ 𝑚) → ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾) ∈ (Base‘𝑅))
77		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (coe₁‘⦋𝑎 / 𝑥⦌𝑀) = (coe₁‘⦋𝑎 / 𝑥⦌𝑀)
78	77, 7, 10, 58	coe1f 21726	. . . . . . . . . . . . . . 15 ⊢ (⦋𝑎 / 𝑥⦌𝑀 ∈ 𝐵 → (coe₁‘⦋𝑎 / 𝑥⦌𝑀):ℕ₀⟶(Base‘𝑅))
79	29, 78	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (coe₁‘⦋𝑎 / 𝑥⦌𝑀):ℕ₀⟶(Base‘𝑅))
80	79, 46	ffvelcdmd 7084	. . . . . . . . . . . . 13 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾) ∈ (Base‘𝑅))
81		nfcv 2903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝑎
82		nfcv 2903	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥coe₁
83		nfcsb1v 3917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝑀
84	82, 83	nffv 6898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(coe₁‘⦋𝑎 / 𝑥⦌𝑀)
85		nfcv 2903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝐾
86	84, 85	nffv 6898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)
87		csbeq1a 3906	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → 𝑀 = ⦋𝑎 / 𝑥⦌𝑀)
88	87	fveq2d 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑎 → (coe₁‘𝑀) = (coe₁‘⦋𝑎 / 𝑥⦌𝑀))
89	88	fveq1d 6890	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑎 → ((coe₁‘𝑀)‘𝐾) = ((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾))
90	81, 86, 89	csbhypf 3921	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾) = ((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾))
91	58, 47, 62, 17, 76, 24, 25, 80, 90	gsumunsn 19822	. . . . . . . . . . . 12 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑅 Σ_g (𝑦 ∈ (𝑚 ∪ {𝑎}) ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾))) = ((𝑅 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
92	57, 91	eqtrid 2784	. . . . . . . . . . 11 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))) = ((𝑅 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)))
93	53, 54, 55	cbvmpt 5258	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)) = (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾))
94	93	eqcomi 2741	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾)) = (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))
95	94	oveq2i 7416	. . . . . . . . . . . 12 ⊢ (𝑅 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾))) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))
96	95	oveq1i 7415	. . . . . . . . . . 11 ⊢ ((𝑅 Σ_g (𝑦 ∈ 𝑚 ↦ ⦋𝑦 / 𝑥⦌((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)) = ((𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾))
97	92, 96	eqtr2di 2789	. . . . . . . . . 10 ⊢ ((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ((𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))
98	97	adantr 481	. . . . . . . . 9 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))) → ((𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))(+_g‘𝑅)((coe₁‘⦋𝑎 / 𝑥⦌𝑀)‘𝐾)) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))
99	52, 98	eqtrd 2772	. . . . . . . 8 ⊢ (((((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) ∧ ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))) → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))
100	99	exp31 420	. . . . . . 7 ⊢ (((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → (∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵 → (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))) → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))))
101	100	com23 86	. . . . . 6 ⊢ (((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ ∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵) → (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))) → (∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))))
102	101	ex 413	. . . . 5 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → (((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))) → (∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾)))))))
103	102	a2d 29	. . . 4 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))) → (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → (∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾)))))))
104	103	imp4b 422	. . 3 ⊢ (((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))))) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 ∧ ∀𝑥 ∈ {𝑎}𝑀 ∈ 𝐵) → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾)))))
105	1, 104	biimtrid 241	. 2 ⊢ (((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) ∧ (∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾))))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾)))))
106	105	ex 413	1 ⊢ ((𝑚 ∈ Fin ∧ ¬ 𝑎 ∈ 𝑚 ∧ 𝜑) → ((∀𝑥 ∈ 𝑚 𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ 𝑚 ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ 𝑚 ↦ ((coe₁‘𝑀)‘𝐾)))) → (∀𝑥 ∈ (𝑚 ∪ {𝑎})𝑀 ∈ 𝐵 → ((coe₁‘(𝑃 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ 𝑀)))‘𝐾) = (𝑅 Σ_g (𝑥 ∈ (𝑚 ∪ {𝑎}) ↦ ((coe₁‘𝑀)‘𝐾))))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⦋csb 3892 ∪ cun 3945 {csn 4627 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 Fincfn 8935 ℕ₀cn0 12468 Basecbs 17140 +_gcplusg 17193 Σ_g cgsu 17382 CMndccmn 19642 Ringcrg 20049 Poly₁cpl1 21692 coe₁cco1 21693

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-psr 21453 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-ply1 21697 df-coe1 21698

This theorem is referenced by: coe1fzgsumd 21817

W3C validator