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Theorem rngcsect 44446
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 (𝜑𝑈𝑉)
rngcsect.x (𝜑𝑋𝐵)
rngcsect.y (𝜑𝑌𝐵)
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) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))

Proof of Theorem rngcsect
StepHypRef Expression
1 rngcsect.b . . 3 𝐵 = (Base‘𝐶)
2 eqid 2824 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2824 . . 3 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2824 . . 3 (Id‘𝐶) = (Id‘𝐶)
5 rngcsect.n . . 3 𝑆 = (Sect‘𝐶)
6 rngcsect.u . . . 4 (𝜑𝑈𝑉)
7 rngcsect.c . . . . 5 𝐶 = (RngCat‘𝑈)
87rngccat 44444 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
96, 8syl 17 . . 3 (𝜑𝐶 ∈ Cat)
10 rngcsect.x . . 3 (𝜑𝑋𝐵)
11 rngcsect.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 5, 9, 10, 11issect 17012 . 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))))
137, 1, 6, 2, 10, 11rngchom 44433 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RngHomo 𝑌))
1413eleq2d 2901 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RngHomo 𝑌)))
157, 1, 6, 2, 11, 10rngchom 44433 . . . . . . 7 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RngHomo 𝑋))
1615eleq2d 2901 . . . . . 6 (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ↔ 𝐺 ∈ (𝑌 RngHomo 𝑋)))
1714, 16anbi12d 633 . . . . 5 (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))))
1817anbi1d 632 . . . 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))))
196adantr 484 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑈𝑉)
2010adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑋𝐵)
217, 1, 6rngcbas 44431 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Rng))
2221eleq2d 2901 . . . . . . . . . 10 (𝜑 → (𝑋𝐵𝑋 ∈ (𝑈 ∩ Rng)))
23 inss1 4188 . . . . . . . . . . . 12 (𝑈 ∩ Rng) ⊆ 𝑈
2423a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑈 ∩ Rng) ⊆ 𝑈)
2524sseld 3950 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ (𝑈 ∩ Rng) → 𝑋𝑈))
2622, 25sylbid 243 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
2726adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝑋𝐵𝑋𝑈))
2820, 27mpd 15 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑋𝑈)
2911adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑌𝐵)
3021eleq2d 2901 . . . . . . . . . 10 (𝜑 → (𝑌𝐵𝑌 ∈ (𝑈 ∩ Rng)))
3124sseld 3950 . . . . . . . . . 10 (𝜑 → (𝑌 ∈ (𝑈 ∩ Rng) → 𝑌𝑈))
3230, 31sylbid 243 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
3332adantr 484 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝑌𝐵𝑌𝑈))
3429, 33mpd 15 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝑌𝑈)
35 eqid 2824 . . . . . . . . . 10 (Base‘𝑋) = (Base‘𝑋)
36 eqid 2824 . . . . . . . . . 10 (Base‘𝑌) = (Base‘𝑌)
3735, 36rnghmf 44365 . . . . . . . . 9 (𝐹 ∈ (𝑋 RngHomo 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
3837adantr 484 . . . . . . . 8 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
3938adantl 485 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
4036, 35rnghmf 44365 . . . . . . . . 9 (𝐺 ∈ (𝑌 RngHomo 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
4140adantl 485 . . . . . . . 8 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
4241adantl 485 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
437, 19, 3, 28, 34, 28, 39, 42rngcco 44437 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝐺𝐹))
44 rngcsect.e . . . . . . . 8 𝐸 = (Base‘𝑋)
457, 1, 4, 6, 10, 44rngcid 44445 . . . . . . 7 (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
4645adantr 484 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
4743, 46eqeq12d 2840 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ↔ (𝐺𝐹) = ( I ↾ 𝐸)))
4847pm5.32da 582 . . . 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)) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
4918, 48bitrd 282 . . 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)) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
50 df-3an 1086 . . 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 1086 . . 3 ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸)) ↔ ((𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋)) ∧ (𝐺𝐹) = ( I ↾ 𝐸)))
5249, 50, 513bitr4g 317 . 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) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
5312, 52bitrd 282 1 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cin 3917  wss 3918  cop 4554   class class class wbr 5047   I cid 5440  cres 5538  ccom 5540  wf 6332  cfv 6336  (class class class)co 7138  Basecbs 16472  Hom chom 16565  compcco 16566  Catccat 16924  Idccid 16925  Sectcsect 17003  Rngcrng 44340   RngHomo crngh 44351  RngCatcrngc 44423
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-3 11687  df-4 11688  df-5 11689  df-6 11690  df-7 11691  df-8 11692  df-9 11693  df-n0 11884  df-z 11968  df-dec 12085  df-uz 12230  df-fz 12884  df-struct 16474  df-ndx 16475  df-slot 16476  df-base 16478  df-sets 16479  df-ress 16480  df-plusg 16567  df-hom 16578  df-cco 16579  df-0g 16704  df-cat 16928  df-cid 16929  df-homf 16930  df-sect 17006  df-ssc 17069  df-resc 17070  df-subc 17071  df-estrc 17362  df-mgm 17841  df-sgrp 17890  df-mnd 17901  df-mhm 17945  df-grp 18095  df-ghm 18345  df-abl 18898  df-mgp 19229  df-mgmhm 44241  df-rng0 44341  df-rnghomo 44353  df-rngc 44425
This theorem is referenced by:  rngcinv  44447
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