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Theorem psropprmul 21751

Description: Reversing multiplication in a ring reverses multiplication in the power series ring. (Contributed by Stefan O'Rear, 27-Mar-2015.)

**Hypotheses**
Ref	Expression
psropprmul.y	⊢ 𝑌 = (𝐼 mPwSer 𝑅)
psropprmul.s	⊢ 𝑆 = (opp_r‘𝑅)
psropprmul.z	⊢ 𝑍 = (𝐼 mPwSer 𝑆)
psropprmul.t	⊢ · = (._r‘𝑌)
psropprmul.u	⊢ ∙ = (._r‘𝑍)
psropprmul.b	⊢ 𝐵 = (Base‘𝑌)

**Assertion**
Ref	Expression
psropprmul	⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Proof of Theorem psropprmul Dummy variables 𝑏 𝑐 𝑒 𝑓 𝑎 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2732	. . . . 5 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
3		ringcmn 20092	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
4	3	3ad2ant1 1133	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝑅 ∈ CMnd)
5	4	adantr 481	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → 𝑅 ∈ CMnd)
6		ovex 7438	. . . . . . . 8 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
7	6	rabex 5331	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 5331	. . . . . 6 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ∈ V
9	8	a1i 11	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ∈ V)
10		simpll1 1212	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → 𝑅 ∈ Ring)
11		psropprmul.y	. . . . . . . . . 10 ⊢ 𝑌 = (𝐼 mPwSer 𝑅)
12		eqid 2732	. . . . . . . . . 10 ⊢ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}
13		psropprmul.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑌)
14		simp3 1138	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ 𝐵)
15	11, 1, 12, 13, 14	psrelbas 21489	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺:{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
16	15	adantr 481	. . . . . . . 8 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → 𝐺:{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
17		elrabi 3676	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
18		ffvelcdm 7080	. . . . . . . 8 ⊢ ((𝐺:{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅) ∧ 𝑒 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝐺‘𝑒) ∈ (Base‘𝑅))
19	16, 17, 18	syl2an 596	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝐺‘𝑒) ∈ (Base‘𝑅))
20		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ 𝐵)
21	11, 1, 12, 13, 20	psrelbas 21489	. . . . . . . . 9 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹:{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
22	21	ad2antrr 724	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → 𝐹:{𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
23		ssrab2 4076	. . . . . . . . 9 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ⊆ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}
24		eqid 2732	. . . . . . . . . . 11 ⊢ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} = {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}
25	12, 24	psrbagconcl 21478	. . . . . . . . . 10 ⊢ ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝑏 ∘_f − 𝑒) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
26	25	adantll 712	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝑏 ∘_f − 𝑒) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
27	23, 26	sselid 3979	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝑏 ∘_f − 𝑒) ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
28	22, 27	ffvelcdmd 7084	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝐹‘(𝑏 ∘_f − 𝑒)) ∈ (Base‘𝑅))
29		eqid 2732	. . . . . . . 8 ⊢ (._r‘𝑅) = (._r‘𝑅)
30	1, 29	ringcl 20066	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝐺‘𝑒) ∈ (Base‘𝑅) ∧ (𝐹‘(𝑏 ∘_f − 𝑒)) ∈ (Base‘𝑅)) → ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))) ∈ (Base‘𝑅))
31	10, 19, 28, 30	syl3anc 1371	. . . . . 6 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))) ∈ (Base‘𝑅))
32	31	fmpttd 7111	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))):{𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}⟶(Base‘𝑅))
33		mptexg 7219	. . . . . . 7 ⊢ ({𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ∈ V → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∈ V)
34	8, 33	mp1i 13	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∈ V)
35		funmpt 6583	. . . . . . 7 ⊢ Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))))
36	35	a1i 11	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))))
37		fvexd 6903	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (0_g‘𝑅) ∈ V)
38	12	psrbaglefi 21476	. . . . . . 7 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} → {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ∈ Fin)
39	38	adantl 482	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ∈ Fin)
40		suppssdm 8158	. . . . . . . 8 ⊢ ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) supp (0_g‘𝑅)) ⊆ dom (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))))
41		eqid 2732	. . . . . . . . 9 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) = (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))))
42	41	dmmptss 6237	. . . . . . . 8 ⊢ dom (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}
43	40, 42	sstri 3990	. . . . . . 7 ⊢ ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) supp (0_g‘𝑅)) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}
44	43	a1i 11	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) supp (0_g‘𝑅)) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
45		suppssfifsupp 9374	. . . . . 6 ⊢ ((((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∈ V ∧ Fun (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∧ (0_g‘𝑅) ∈ V) ∧ ({𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ∈ Fin ∧ ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) supp (0_g‘𝑅)) ⊆ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) finSupp (0_g‘𝑅))
46	34, 36, 37, 39, 44, 45	syl32anc 1378	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) finSupp (0_g‘𝑅))
47	12, 24	psrbagconf1o 21480	. . . . . 6 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐)):{𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}–1-1-onto→{𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
48	47	adantl 482	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐)):{𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}–1-1-onto→{𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
49	1, 2, 5, 9, 32, 46, 48	gsumf1o 19778	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σ_g (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))))) = (𝑅 Σ_g ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐)))))
50	12, 24	psrbagconcl 21478	. . . . . . . 8 ⊢ ((𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝑏 ∘_f − 𝑐) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
51	50	adantll 712	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝑏 ∘_f − 𝑐) ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏})
52		eqidd 2733	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐)) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐)))
53		eqidd 2733	. . . . . . 7 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) = (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))))
54		fveq2 6888	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘_f − 𝑐)))
55		oveq2 7413	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → (𝑏 ∘_f − 𝑒) = (𝑏 ∘_f − (𝑏 ∘_f − 𝑐)))
56	55	fveq2d 6892	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → (𝐹‘(𝑏 ∘_f − 𝑒)) = (𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐))))
57	54, 56	oveq12d 7423	. . . . . . 7 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))) = ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐)))))
58	51, 52, 53, 57	fmptco 7123	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐))) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐))))))
59		reldmpsr 21458	. . . . . . . . . . . . . 14 ⊢ Rel dom mPwSer
60	11, 13, 59	strov2rcl 17148	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ 𝐵 → 𝐼 ∈ V)
61	60	3ad2ant3 1135	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐼 ∈ V)
62	61	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → 𝐼 ∈ V)
63	12	psrbagf 21462	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} → 𝑏:𝐼⟶ℕ₀)
64	63	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → 𝑏:𝐼⟶ℕ₀)
65	64	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → 𝑏:𝐼⟶ℕ₀)
66		elrabi 3676	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
67	12	psrbagf 21462	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} → 𝑐:𝐼⟶ℕ₀)
68	66, 67	syl 17	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} → 𝑐:𝐼⟶ℕ₀)
69	68	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → 𝑐:𝐼⟶ℕ₀)
70		nn0cn 12478	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ₀ → 𝑒 ∈ ℂ)
71		nn0cn 12478	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ₀ → 𝑓 ∈ ℂ)
72		nncan 11485	. . . . . . . . . . . . 13 ⊢ ((𝑒 ∈ ℂ ∧ 𝑓 ∈ ℂ) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
73	70, 71, 72	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑒 ∈ ℕ₀ ∧ 𝑓 ∈ ℕ₀) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
74	73	adantl 482	. . . . . . . . . . 11 ⊢ (((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) ∧ (𝑒 ∈ ℕ₀ ∧ 𝑓 ∈ ℕ₀)) → (𝑒 − (𝑒 − 𝑓)) = 𝑓)
75	62, 65, 69, 74	caonncan 7707	. . . . . . . . . 10 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝑏 ∘_f − (𝑏 ∘_f − 𝑐)) = 𝑐)
76	75	fveq2d 6892	. . . . . . . . 9 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → (𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐))) = (𝐹‘𝑐))
77	76	oveq2d 7421	. . . . . . . 8 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐)))) = ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘𝑐)))
78		psropprmul.s	. . . . . . . . 9 ⊢ 𝑆 = (opp_r‘𝑅)
79		eqid 2732	. . . . . . . . 9 ⊢ (._r‘𝑆) = (._r‘𝑆)
80	1, 29, 78, 79	opprmul 20145	. . . . . . . 8 ⊢ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))) = ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘𝑐))
81	77, 80	eqtr4di 2790	. . . . . . 7 ⊢ ((((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) ∧ 𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏}) → ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐)))) = ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))
82	81	mpteq2dva 5247	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘(𝑏 ∘_f − 𝑐))(._r‘𝑅)(𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐))))) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐)))))
83	58, 82	eqtrd 2772	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐))) = (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐)))))
84	83	oveq2d 7421	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σ_g ((𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))) ∘ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ (𝑏 ∘_f − 𝑐)))) = (𝑅 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))))
85	8	mptex 7221	. . . . . . . 8 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐)))) ∈ V
86	85	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐)))) ∈ V)
87		id 22	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Ring)
88	78	fvexi 6902	. . . . . . . 8 ⊢ 𝑆 ∈ V
89	88	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
90	78, 1	opprbas 20149	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑆)
91	90	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (Base‘𝑅) = (Base‘𝑆))
92		eqid 2732	. . . . . . . . 9 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
93	78, 92	oppradd 20151	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑆)
94	93	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+_g‘𝑅) = (+_g‘𝑆))
95	86, 87, 89, 91, 94	gsumpropd 18593	. . . . . 6 ⊢ (𝑅 ∈ Ring → (𝑅 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))) = (𝑆 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))))
96	95	3ad2ant1 1133	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑅 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))) = (𝑆 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))))
97	96	adantr 481	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))) = (𝑆 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))))
98	49, 84, 97	3eqtrd 2776	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (𝑅 Σ_g (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒))))) = (𝑆 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐))))))
99	98	mpteq2dva 5247	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑅 Σ_g (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))))) = (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑆 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐)))))))
100		psropprmul.t	. . 3 ⊢ · = (._r‘𝑌)
101	11, 13, 29, 100, 12, 14, 20	psrmulfval 21495	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐺 · 𝐹) = (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑅 Σ_g (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐺‘𝑒)(._r‘𝑅)(𝐹‘(𝑏 ∘_f − 𝑒)))))))
102		psropprmul.z	. . 3 ⊢ 𝑍 = (𝐼 mPwSer 𝑆)
103		eqid 2732	. . 3 ⊢ (Base‘𝑍) = (Base‘𝑍)
104		psropprmul.u	. . 3 ⊢ ∙ = (._r‘𝑍)
105	90	a1i 11	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘𝑅) = (Base‘𝑆))
106	105	psrbaspropd 21748	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
107	11	fveq2i 6891	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
108	13, 107	eqtri 2760	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
109	102	fveq2i 6891	. . . . 5 ⊢ (Base‘𝑍) = (Base‘(𝐼 mPwSer 𝑆))
110	106, 108, 109	3eqtr4g 2797	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐵 = (Base‘𝑍))
111	20, 110	eleqtrd 2835	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐹 ∈ (Base‘𝑍))
112	14, 110	eleqtrd 2835	. . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → 𝐺 ∈ (Base‘𝑍))
113	102, 103, 79, 104, 12, 111, 112	psrmulfval 21495	. 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ↦ (𝑆 Σ_g (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} ↦ ((𝐹‘𝑐)(._r‘𝑆)(𝐺‘(𝑏 ∘_f − 𝑐)))))))
114	99, 101, 113	3eqtr4rd 2783	1 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (𝐹 ∙ 𝐺) = (𝐺 · 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ^◡ccnv 5674 dom cdm 5675 “ cima 5678 ∘ ccom 5679 Fun wfun 6534 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 ∘_r cofr 7665 supp csupp 8142 ↑_m cmap 8816 Fincfn 8935 finSupp cfsupp 9357 ℂcc 11104 ≤ cle 11245 − cmin 11440 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 +_gcplusg 17193 ._rcmulr 17194 0_gc0g 17381 Σ_g cgsu 17382 CMndccmn 19642 Ringcrg 20049 opp_rcoppr 20141 mPwSer cmps 21448

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-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-tpos 8207 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-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-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-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-tset 17212 df-0g 17383 df-gsum 17384 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-psr 21453

This theorem is referenced by: ply1opprmul 21752

W3C validator