Theorem psrcom 19409

Description: Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)

Hypotheses

Ref	Expression
psrring.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrring.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
psrring.r	⊢ (𝜑 → 𝑅 ∈ Ring)
psrass.d	⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
psrass.t	⊢ × = (.r‘𝑆)
psrass.b	⊢ 𝐵 = (Base‘𝑆)
psrass.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
psrass.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
psrcom.c	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
psrcom	⊢ (𝜑 → (𝑋 × 𝑌) = (𝑌 × 𝑋))

Distinct variable groups:   𝑓,𝐼   𝑅,𝑓   𝑓,𝑋   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑓)   𝐷(𝑓)   𝑆(𝑓)   × (𝑓)   𝑉(𝑓)

Proof of Theorem psrcom

Dummy variables 𝑥 𝑘 𝑧 𝑔 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2622	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		psrring.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		ringcmn 18581	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑅 ∈ CMnd)
7		psrring.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
8		psrass.d	. . . . . . 7 ⊢ 𝐷 = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
9	8	psrbaglefi 19372	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ Fin)
10	7, 9	sylan 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ Fin)
11	3	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑅 ∈ Ring)
12		psrring.s	. . . . . . . . . 10 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
13		psrass.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑆)
14		psrass.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
15	12, 1, 8, 13, 14	psrelbas 19379	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:𝐷⟶(Base‘𝑅))
16	15	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
17		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
18		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑘 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑘 ∘𝑟 ≤ 𝑥))
19	18	elrab 3363	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↔ (𝑘 ∈ 𝐷 ∧ 𝑘 ∘𝑟 ≤ 𝑥))
20	17, 19	sylib 208	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑘 ∈ 𝐷 ∧ 𝑘 ∘𝑟 ≤ 𝑥))
21	20	simpld 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ 𝐷)
22	16, 21	ffvelrnd 6360	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑋‘𝑘) ∈ (Base‘𝑅))
23		psrass.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
24	12, 1, 8, 13, 23	psrelbas 19379	. . . . . . . . 9 ⊢ (𝜑 → 𝑌:𝐷⟶(Base‘𝑅))
25	24	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑌:𝐷⟶(Base‘𝑅))
26	7	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝐼 ∈ 𝑉)
27		simplr 792	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
28	8	psrbagf 19365	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0)
29	26, 21, 28	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
30	20	simprd 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∘𝑟 ≤ 𝑥)
31	8	psrbagcon 19371	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑘:𝐼⟶ℕ0 ∧ 𝑘 ∘𝑟 ≤ 𝑥)) → ((𝑥 ∘𝑓 − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑘) ∘𝑟 ≤ 𝑥))
32	26, 27, 29, 30, 31	syl13anc 1328	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑘) ∘𝑟 ≤ 𝑥))
33	32	simpld 475	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑘) ∈ 𝐷)
34	25, 33	ffvelrnd 6360	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑌‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅))
35		eqid 2622	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	1, 35	ringcl 18561	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋‘𝑘) ∈ (Base‘𝑅) ∧ (𝑌‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅)) → ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))) ∈ (Base‘𝑅))
37	11, 22, 34, 36	syl3anc 1326	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))) ∈ (Base‘𝑅))
38		eqid 2622	. . . . . 6 ⊢ (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) = (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))))
39	37, 38	fmptd 6385	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))):{𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}⟶(Base‘𝑅))
40		ovex 6678	. . . . . . . . . 10 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
41	8, 40	rabex2 4815	. . . . . . . . 9 ⊢ 𝐷 ∈ V
42	41	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐷 ∈ V)
43		rabexg 4812	. . . . . . . 8 ⊢ (𝐷 ∈ V → {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ V)
44	42, 43	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ V)
45		mptexg 6484	. . . . . . 7 ⊢ ({𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ V → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∈ V)
46	44, 45	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∈ V)
47		funmpt 5926	. . . . . . 7 ⊢ Fun (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))))
48	47	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → Fun (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))))
49		fvexd 6203	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (0g‘𝑅) ∈ V)
50		suppssdm 7308	. . . . . . . 8 ⊢ ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) supp (0g‘𝑅)) ⊆ dom (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))))
51	38	dmmptss 5631	. . . . . . . 8 ⊢ dom (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ⊆ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}
52	50, 51	sstri 3612	. . . . . . 7 ⊢ ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) supp (0g‘𝑅)) ⊆ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}
53	52	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) supp (0g‘𝑅)) ⊆ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
54		suppssfifsupp 8290	. . . . . 6 ⊢ ((((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∈ V ∧ Fun (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∧ (0g‘𝑅) ∈ V) ∧ ({𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ Fin ∧ ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) supp (0g‘𝑅)) ⊆ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) finSupp (0g‘𝑅))
55	46, 48, 49, 10, 53, 54	syl32anc 1334	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) finSupp (0g‘𝑅))
56		eqid 2622	. . . . . . 7 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}
57	8, 56	psrbagconf1o 19374	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)):{𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}–1-1-onto→{𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
58	7, 57	sylan 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)):{𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}–1-1-onto→{𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
59	1, 2, 6, 10, 39, 55, 58	gsumf1o 18317	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))))) = (𝑅 Σg ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∘ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)))))
60	7	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝐼 ∈ 𝑉)
61		simplr 792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
62		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
63	8, 56	psrbagconcl 19373	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
64	60, 61, 62, 63	syl3anc 1326	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
65		eqidd 2623	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)))
66		eqidd 2623	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) = (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))))
67		fveq2 6191	. . . . . . . 8 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → (𝑋‘𝑘) = (𝑋‘(𝑥 ∘𝑓 − 𝑗)))
68		oveq2 6658	. . . . . . . . 9 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → (𝑥 ∘𝑓 − 𝑘) = (𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)))
69	68	fveq2d 6195	. . . . . . . 8 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → (𝑌‘(𝑥 ∘𝑓 − 𝑘)) = (𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗))))
70	67, 69	oveq12d 6668	. . . . . . 7 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))) = ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)))))
71	64, 65, 66, 70	fmptco 6396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∘ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗))))))
72	8	psrbagf 19365	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
73	7, 72	sylan 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
74	73	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥:𝐼⟶ℕ0)
75	74	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
76		breq1 4656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑗 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑗 ∘𝑟 ≤ 𝑥))
77	76	elrab 3363	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↔ (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑥))
78	62, 77	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑗 ∈ 𝐷 ∧ 𝑗 ∘𝑟 ≤ 𝑥))
79	78	simpld 475	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ 𝐷)
80	8	psrbagf 19365	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐷) → 𝑗:𝐼⟶ℕ0)
81	60, 79, 80	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
82	81	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
83		nn0cn 11302	. . . . . . . . . . . . . 14 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
84		nn0cn 11302	. . . . . . . . . . . . . 14 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
85		nncan 10310	. . . . . . . . . . . . . 14 ⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧))) = (𝑗‘𝑧))
86	83, 84, 85	syl2an 494	. . . . . . . . . . . . 13 ⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧))) = (𝑗‘𝑧))
87	75, 82, 86	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧))) = (𝑗‘𝑧))
88	87	mpteq2dva 4744	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
89		ovex 6678	. . . . . . . . . . . . 13 ⊢ ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V
90	89	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − (𝑗‘𝑧)) ∈ V)
91	74	feqmptd 6249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
92	81	feqmptd 6249	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
93	60, 75, 82, 91, 92	offval2 6914	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − (𝑗‘𝑧))))
94	60, 75, 90, 91, 93	offval2 6914	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)) = (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧)))))
95	88, 94, 92	3eqtr4d 2666	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)) = 𝑗)
96	95	fveq2d 6195	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗))) = (𝑌‘𝑗))
97	96	oveq2d 6666	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)))) = ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘𝑗)))
98		psrcom.c	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ CRing)
99	98	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑅 ∈ CRing)
100	15	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑋:𝐷⟶(Base‘𝑅))
101	78	simprd 479	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∘𝑟 ≤ 𝑥)
102	8	psrbagcon 19371	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝑥 ∈ 𝐷 ∧ 𝑗:𝐼⟶ℕ0 ∧ 𝑗 ∘𝑟 ≤ 𝑥)) → ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑥))
103	60, 61, 81, 101, 102	syl13anc 1328	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑥))
104	103	simpld 475	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷)
105	100, 104	ffvelrnd 6360	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑋‘(𝑥 ∘𝑓 − 𝑗)) ∈ (Base‘𝑅))
106	24	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑌:𝐷⟶(Base‘𝑅))
107	106, 79	ffvelrnd 6360	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑌‘𝑗) ∈ (Base‘𝑅))
108	1, 35	crngcom 18562	. . . . . . . . 9 ⊢ ((𝑅 ∈ CRing ∧ (𝑋‘(𝑥 ∘𝑓 − 𝑗)) ∈ (Base‘𝑅) ∧ (𝑌‘𝑗) ∈ (Base‘𝑅)) → ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘𝑗)) = ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗))))
109	99, 105, 107, 108	syl3anc 1326	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘𝑗)) = ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗))))
110	97, 109	eqtrd 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)))) = ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗))))
111	110	mpteq2dva 4744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘(𝑥 ∘𝑓 − 𝑗))(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗))))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗)))))
112	71, 111	eqtrd 2656	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∘ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗))) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗)))))
113	112	oveq2d 6666	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg ((𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) ∘ (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗))))))
114	59, 113	eqtrd 2656	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))))) = (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗))))))
115	114	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))))) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗)))))))
116		psrass.t	. . 3 ⊢ × = (.r‘𝑆)
117	12, 13, 35, 116, 8, 14, 23	psrmulfval 19385	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))))))
118	12, 13, 35, 116, 8, 23, 14	psrmulfval 19385	. 2 ⊢ (𝜑 → (𝑌 × 𝑋) = (𝑥 ∈ 𝐷 ↦ (𝑅 Σg (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑌‘𝑗)(.r‘𝑅)(𝑋‘(𝑥 ∘𝑓 − 𝑗)))))))
119	115, 117, 118	3eqtr4d 2666	1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑌 × 𝑋))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ⊆ wss 3574   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  dom cdm 5114   “ cima 5117   ∘ ccom 5118  Fun wfun 5882  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ∘_𝑟 cofr 6896   supp csupp 7295   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275  ℂcc 9934   ≤ cle 10075   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857  ._rcmulr 15942  0_gc0g 16100   Σ_g cgsu 16101  CMndccmn 18193  Ringcrg 18547  CRingccrg 18548   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-psr 19356

This theorem is referenced by:  psrcrng  19413