Theorem rngcsect 45538

Description: A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.)

Hypotheses

Ref	Expression
rngcsect.c	⊢ 𝐶 = (RngCat‘𝑈)
rngcsect.b	⊢ 𝐵 = (Base‘𝐶)
rngcsect.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
rngcsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
rngcsect.e	⊢ 𝐸 = (Base‘𝑋)
rngcsect.n	⊢ 𝑆 = (Sect‘𝐶)

Assertion

Ref	Expression
rngcsect	⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))

Proof of Theorem rngcsect

Step	Hyp	Ref	Expression
1		rngcsect.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2738	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2738	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2738	. . 3 ⊢ (Id‘𝐶) = (Id‘𝐶)
5		rngcsect.n	. . 3 ⊢ 𝑆 = (Sect‘𝐶)
6		rngcsect.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
7		rngcsect.c	. . . . 5 ⊢ 𝐶 = (RngCat‘𝑈)
8	7	rngccat 45536	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
9	6, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
10		rngcsect.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		rngcsect.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
12	1, 2, 3, 4, 5, 9, 10, 11	issect 17465	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
13	7, 1, 6, 2, 10, 11	rngchom 45525	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RngHomo 𝑌))
14	13	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌)))
15	7, 1, 6, 2, 11, 10	rngchom 45525	. . . . . . 7 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RngHomo 𝑋))
16	15	eleq2d 2824	. . . . . 6 ⊢ (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
17	14, 16	anbi12d 631	. . . . 5 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))))
18	17	anbi1d 630	. . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
19	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑈 ∈ 𝑉)
20	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑋 ∈ 𝐵)
21	7, 1, 6	rngcbas 45523	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
22	21	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Rng)))
23		inss1 4162	. . . . . . . . . . . 12 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈)
25	24	sseld 3920	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ (𝑈 ∩ Rng) → 𝑋 ∈ 𝑈))
26	22, 25	sylbid 239	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈))
27	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝑈))
28	20, 27	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑋 ∈ 𝑈)
29	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑌 ∈ 𝐵)
30	21	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝑈 ∩ Rng)))
31	24	sseld 3920	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ (𝑈 ∩ Rng) → 𝑌 ∈ 𝑈))
32	30, 31	sylbid 239	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈))
33	32	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈))
34	29, 33	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑌 ∈ 𝑈)
35		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑋) = (Base‘𝑋)
36		eqid 2738	. . . . . . . . . 10 ⊢ (Base‘𝑌) = (Base‘𝑌)
37	35, 36	rnghmf 45457	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑋 RngHomo 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
38	37	adantr 481	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
39	38	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
40	36, 35	rnghmf 45457	. . . . . . . . 9 ⊢ (𝐺 ∈ (𝑌 RngHomo 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
41	40	adantl 482	. . . . . . . 8 ⊢ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
42	41	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
43	7, 19, 3, 28, 34, 28, 39, 42	rngcco 45529	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝐺 ∘ 𝐹))
44		rngcsect.e	. . . . . . . 8 ⊢ 𝐸 = (Base‘𝑋)
45	7, 1, 4, 6, 10, 44	rngcid 45537	. . . . . . 7 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
46	45	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
47	43, 46	eqeq12d 2754	. . . . 5 ⊢ ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ↔ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))
48	47	pm5.32da 579	. . . 4 ⊢ (𝜑 → (((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
49	18, 48	bitrd 278	. . 3 ⊢ (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
50		df-3an 1088	. . 3 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)))
51		df-3an 1088	. . 3 ⊢ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))
52	49, 50, 51	3bitr4g 314	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
53	12, 52	bitrd 278	1 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887  ⟨cop 4567   class class class wbr 5074   I cid 5488   ↾ cres 5591   ∘ ccom 5593  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  Idccid 17374  Sectcsect 17456  Rngcrng 45432   RngHomo crngh 45443  RngCatcrngc 45515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-pm 8618  df-ixp 8686  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-hom 16986  df-cco 16987  df-0g 17152  df-cat 17377  df-cid 17378  df-homf 17379  df-sect 17459  df-ssc 17522  df-resc 17523  df-subc 17524  df-estrc 17839  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-grp 18580  df-ghm 18832  df-abl 19389  df-mgp 19721  df-mgmhm 45333  df-rng0 45433  df-rnghomo 45445  df-rngc 45517

This theorem is referenced by:  rngcinv  45539