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Theorem psropprmul 21767

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 20101	. . . . . . 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 7444	. . . . . . . 8 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
7	6	rabex 5332	. . . . . . 7 ⊢ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∈ V
8	7	rabex 5332	. . . . . 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 21504	. . . . . . . . 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 3677	. . . . . . . 8 ⊢ (𝑒 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} → 𝑒 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
18		ffvelcdm 7083	. . . . . . . 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 21504	. . . . . . . . 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 4077	. . . . . . . . 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 21493	. . . . . . . . . 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 3980	. . . . . . . 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 7087	. . . . . . 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 20075	. . . . . . 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 7116	. . . . 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 7225	. . . . . . 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 6586	. . . . . . 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 6906	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) ∧ 𝑏 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin}) → (0_g‘𝑅) ∈ V)
38	12	psrbaglefi 21491	. . . . . . 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 8164	. . . . . . . 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 6240	. . . . . . . 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 3991	. . . . . . 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 9380	. . . . . 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 21495	. . . . . 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 19786	. . . 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 21493	. . . . . . . 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 6891	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → (𝐺‘𝑒) = (𝐺‘(𝑏 ∘_f − 𝑐)))
55		oveq2 7419	. . . . . . . . 9 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → (𝑏 ∘_f − 𝑒) = (𝑏 ∘_f − (𝑏 ∘_f − 𝑐)))
56	55	fveq2d 6895	. . . . . . . 8 ⊢ (𝑒 = (𝑏 ∘_f − 𝑐) → (𝐹‘(𝑏 ∘_f − 𝑒)) = (𝐹‘(𝑏 ∘_f − (𝑏 ∘_f − 𝑐))))
57	54, 56	oveq12d 7429	. . . . . . 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 7129	. . . . . 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 21473	. . . . . . . . . . . . . 14 ⊢ Rel dom mPwSer
60	11, 13, 59	strov2rcl 17154	. . . . . . . . . . . . 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 21477	. . . . . . . . . . . . 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 3677	. . . . . . . . . . . . 13 ⊢ (𝑐 ∈ {𝑑 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin} ∣ 𝑑 ∘_r ≤ 𝑏} → 𝑐 ∈ {𝑎 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑎 “ ℕ) ∈ Fin})
67	12	psrbagf 21477	. . . . . . . . . . . . 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 12484	. . . . . . . . . . . . 13 ⊢ (𝑒 ∈ ℕ₀ → 𝑒 ∈ ℂ)
71		nn0cn 12484	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ₀ → 𝑓 ∈ ℂ)
72		nncan 11491	. . . . . . . . . . . . 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 7713	. . . . . . . . . 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 6895	. . . . . . . . 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 7427	. . . . . . . 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 20157	. . . . . . . 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 5248	. . . . . 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 7427	. . . 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 7227	. . . . . . . 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 6905	. . . . . . . 8 ⊢ 𝑆 ∈ V
89	88	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑆 ∈ V)
90	78, 1	opprbas 20161	. . . . . . . 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 20163	. . . . . . . 8 ⊢ (+_g‘𝑅) = (+_g‘𝑆)
94	93	a1i 11	. . . . . . 7 ⊢ (𝑅 ∈ Ring → (+_g‘𝑅) = (+_g‘𝑆))
95	86, 87, 89, 91, 94	gsumpropd 18599	. . . . . 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 5248	. 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 21510	. 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 21764	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ 𝐵) → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
107	11	fveq2i 6894	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘(𝐼 mPwSer 𝑅))
108	13, 107	eqtri 2760	. . . . 5 ⊢ 𝐵 = (Base‘(𝐼 mPwSer 𝑅))
109	102	fveq2i 6894	. . . . 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 21510	. 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 3948 class class class wbr 5148 ↦ cmpt 5231 ^◡ccnv 5675 dom cdm 5676 “ cima 5679 ∘ ccom 5680 Fun wfun 6537 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 ∘_f cof 7670 ∘_r cofr 7671 supp csupp 8148 ↑_m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 ℂcc 11110 ≤ cle 11251 − cmin 11446 ℕcn 12214 ℕ₀cn0 12474 Basecbs 17146 +_gcplusg 17199 ._rcmulr 17200 0_gc0g 17387 Σ_g cgsu 17388 CMndccmn 19650 Ringcrg 20058 opp_rcoppr 20153 mPwSer cmps 21463

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-ofr 7673 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-fzo 13630 df-seq 13969 df-hash 14293 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-tset 17218 df-0g 17389 df-gsum 17390 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-oppr 20154 df-psr 21468

This theorem is referenced by: ply1opprmul 21768

W3C validator