Theorem psrcom 19457

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 2651	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
2		eqid 2651	. . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅)
3		psrring.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
4		ringcmn 18627	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
6	5	adantr 480	. . . . 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 19420	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ∈ Fin)
10	7, 9	sylan 487	. . . . 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 19427	. . . . . . . . 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 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
18		breq1 4688	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑘 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑘 ∘𝑟 ≤ 𝑥))
19	18	elrab 3396	. . . . . . . . . 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 474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∈ 𝐷)
22	16, 21	ffvelrnd 6400	. . . . . . 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 19427	. . . . . . . . 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 807	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
28	8	psrbagf 19413	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑘 ∈ 𝐷) → 𝑘:𝐼⟶ℕ0)
29	26, 21, 28	syl2anc 694	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘:𝐼⟶ℕ0)
30	20	simprd 478	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑘 ∘𝑟 ≤ 𝑥)
31	8	psrbagcon 19419	. . . . . . . . . 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 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑘) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑘) ∘𝑟 ≤ 𝑥))
33	32	simpld 474	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑘) ∈ 𝐷)
34	25, 33	ffvelrnd 6400	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑌‘(𝑥 ∘𝑓 − 𝑘)) ∈ (Base‘𝑅))
35		eqid 2651	. . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅)
36	1, 35	ringcl 18607	. . . . . . 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 1366	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘))) ∈ (Base‘𝑅))
38		eqid 2651	. . . . . 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 6425	. . . . 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 6718	. . . . . . . . . 10 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
41	8, 40	rabex2 4847	. . . . . . . . 9 ⊢ 𝐷 ∈ V
42	41	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐷 ∈ V)
43		rabexg 4844	. . . . . . . 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 6525	. . . . . . 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 5964	. . . . . . 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 6241	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (0g‘𝑅) ∈ V)
50		suppssdm 7353	. . . . . . . 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 5669	. . . . . . . 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 3645	. . . . . . 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 8331	. . . . . 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 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑘 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ ((𝑋‘𝑘)(.r‘𝑅)(𝑌‘(𝑥 ∘𝑓 − 𝑘)))) finSupp (0g‘𝑅))
56		eqid 2651	. . . . . . 7 ⊢ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} = {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}
57	8, 56	psrbagconf1o 19422	. . . . . 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 487	. . . . 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 18363	. . . 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 807	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 ∈ 𝐷)
62		simpr 476	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
63	8, 56	psrbagconcl 19421	. . . . . . . 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 1366	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥})
65		eqidd 2652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)) = (𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥} ↦ (𝑥 ∘𝑓 − 𝑗)))
66		eqidd 2652	. . . . . . 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 6229	. . . . . . . 8 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → (𝑋‘𝑘) = (𝑋‘(𝑥 ∘𝑓 − 𝑗)))
68		oveq2 6698	. . . . . . . . 9 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → (𝑥 ∘𝑓 − 𝑘) = (𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)))
69	68	fveq2d 6233	. . . . . . . 8 ⊢ (𝑘 = (𝑥 ∘𝑓 − 𝑗) → (𝑌‘(𝑥 ∘𝑓 − 𝑘)) = (𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗))))
70	67, 69	oveq12d 6708	. . . . . . 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 6436	. . . . . 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 19413	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
73	7, 72	sylan 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥:𝐼⟶ℕ0)
74	73	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥:𝐼⟶ℕ0)
75	74	ffvelrnda 6399	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → (𝑥‘𝑧) ∈ ℕ0)
76		breq1 4688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝑗 → (𝑔 ∘𝑟 ≤ 𝑥 ↔ 𝑗 ∘𝑟 ≤ 𝑥))
77	76	elrab 3396	. . . . . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∈ 𝐷)
80	8	psrbagf 19413	. . . . . . . . . . . . . . 15 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝑗 ∈ 𝐷) → 𝑗:𝐼⟶ℕ0)
81	60, 79, 80	syl2anc 694	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗:𝐼⟶ℕ0)
82	81	ffvelrnda 6399	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → (𝑗‘𝑧) ∈ ℕ0)
83		nn0cn 11340	. . . . . . . . . . . . . 14 ⊢ ((𝑥‘𝑧) ∈ ℕ0 → (𝑥‘𝑧) ∈ ℂ)
84		nn0cn 11340	. . . . . . . . . . . . . 14 ⊢ ((𝑗‘𝑧) ∈ ℕ0 → (𝑗‘𝑧) ∈ ℂ)
85		nncan 10348	. . . . . . . . . . . . . 14 ⊢ (((𝑥‘𝑧) ∈ ℂ ∧ (𝑗‘𝑧) ∈ ℂ) → ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧))) = (𝑗‘𝑧))
86	83, 84, 85	syl2an 493	. . . . . . . . . . . . 13 ⊢ (((𝑥‘𝑧) ∈ ℕ0 ∧ (𝑗‘𝑧) ∈ ℕ0) → ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧))) = (𝑗‘𝑧))
87	75, 82, 86	syl2anc 694	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) ∧ 𝑧 ∈ 𝐼) → ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧))) = (𝑗‘𝑧))
88	87	mpteq2dva 4777	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑧 ∈ 𝐼 ↦ ((𝑥‘𝑧) − ((𝑥‘𝑧) − (𝑗‘𝑧)))) = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
89		ovex 6718	. . . . . . . . . . . . 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 6288	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑥 = (𝑧 ∈ 𝐼 ↦ (𝑥‘𝑧)))
92	81	feqmptd 6288	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 = (𝑧 ∈ 𝐼 ↦ (𝑗‘𝑧)))
93	60, 75, 82, 91, 92	offval2 6956	. . . . . . . . . . . 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 6956	. . . . . . . . . . 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 2695	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗)) = 𝑗)
96	95	fveq2d 6233	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑌‘(𝑥 ∘𝑓 − (𝑥 ∘𝑓 − 𝑗))) = (𝑌‘𝑗))
97	96	oveq2d 6706	. . . . . . . 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 478	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → 𝑗 ∘𝑟 ≤ 𝑥)
102	8	psrbagcon 19419	. . . . . . . . . . . 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 1368	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → ((𝑥 ∘𝑓 − 𝑗) ∈ 𝐷 ∧ (𝑥 ∘𝑓 − 𝑗) ∘𝑟 ≤ 𝑥))
104	103	simpld 474	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑥 ∘𝑓 − 𝑗) ∈ 𝐷)
105	100, 104	ffvelrnd 6400	. . . . . . . . 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 6400	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑗 ∈ {𝑔 ∈ 𝐷 ∣ 𝑔 ∘𝑟 ≤ 𝑥}) → (𝑌‘𝑗) ∈ (Base‘𝑅))
108	1, 35	crngcom 18608	. . . . . . . . 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 1366	. . . . . . . 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 2685	. . . . . . 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 4777	. . . . . 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 2685	. . . . 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 6706	. . . 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 2685	. . 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 4777	. 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 19433	. 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 19433	. 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 2695	1 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑌 × 𝑋))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {crab 2945  Vcvv 3231   ⊆ wss 3607   class class class wbr 4685   ↦ cmpt 4762  ^◡ccnv 5142  dom cdm 5143   “ cima 5146   ∘ ccom 5147  Fun wfun 5920  ⟶wf 5922  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690   ∘_𝑓 cof 6937   ∘_𝑟 cofr 6938   supp csupp 7340   ↑_𝑚 cmap 7899  Fincfn 7997   finSupp cfsupp 8316  ℂcc 9972   ≤ cle 10113   − cmin 10304  ℕcn 11058  ℕ₀cn0 11330  Basecbs 15904  ._rcmulr 15989  0_gc0g 16147   Σ_g cgsu 16148  CMndccmn 18239  Ringcrg 18593  CRingccrg 18594   mPwSer cmps 19399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-psr 19404

This theorem is referenced by:  psrcrng  19461