Theorem rngcsect 46398

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 2731	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2731	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2731	. . 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 46396	. . . 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 17650	. 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 46385	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RngHomo 𝑌))
14	13	eleq2d 2818	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌)))
15	7, 1, 6, 2, 11, 10	rngchom 46385	. . . . . . 7 ⊢ (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RngHomo 𝑋))
16	15	eleq2d 2818	. . . . . 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 46383	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
22	21	eleq2d 2818	. . . . . . . . . 10 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ (𝑈 ∩ Rng)))
23		inss1 4193	. . . . . . . . . . . 12 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
24	23	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈)
25	24	sseld 3946	. . . . . . . . . 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 2818	. . . . . . . . . 10 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝑈 ∩ Rng)))
31	24	sseld 3946	. . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (Base‘𝑋) = (Base‘𝑋)
36		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘𝑌) = (Base‘𝑌)
37	35, 36	rnghmf 46317	. . . . . . . . 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 46317	. . . . . . . . 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 46389	. . . . . 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 46397	. . . . . . 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 2747	. . . . 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 1089	. . 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 1089	. . 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 313	. 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 1087   = wceq 1541   ∈ wcel 2106   ∩ cin 3912   ⊆ wss 3913  ⟨cop 4597   class class class wbr 5110   I cid 5535   ↾ cres 5640   ∘ ccom 5642  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  Hom chom 17158  compcco 17159  Catccat 17558  Idccid 17559  Sectcsect 17641  Rngcrng 46292   RngHomo crngh 46303  RngCatcrngc 46375

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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-pm 8775  df-ixp 8843  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12423  df-z 12509  df-dec 12628  df-uz 12773  df-fz 13435  df-struct 17030  df-sets 17047  df-slot 17065  df-ndx 17077  df-base 17095  df-ress 17124  df-plusg 17160  df-hom 17171  df-cco 17172  df-0g 17337  df-cat 17562  df-cid 17563  df-homf 17564  df-sect 17644  df-ssc 17707  df-resc 17708  df-subc 17709  df-estrc 18024  df-mgm 18511  df-sgrp 18560  df-mnd 18571  df-mhm 18615  df-grp 18765  df-ghm 19020  df-abl 19579  df-mgp 19911  df-mgmhm 46193  df-rng 46293  df-rnghomo 46305  df-rngc 46377

This theorem is referenced by:  rngcinv  46399