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Theorem ringcsect 42356
 Description: A section in the category of unital rings, written out. (Contributed by AV, 14-Feb-2020.)
Hypotheses
Ref Expression
ringcsect.c 𝐶 = (RingCat‘𝑈)
ringcsect.b 𝐵 = (Base‘𝐶)
ringcsect.u (𝜑𝑈𝑉)
ringcsect.x (𝜑𝑋𝐵)
ringcsect.y (𝜑𝑌𝐵)
ringcsect.e 𝐸 = (Base‘𝑋)
ringcsect.n 𝑆 = (Sect‘𝐶)
Assertion
Ref Expression
ringcsect (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))

Proof of Theorem ringcsect
StepHypRef Expression
1 ringcsect.b . . 3 𝐵 = (Base‘𝐶)
2 eqid 2651 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2651 . . 3 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2651 . . 3 (Id‘𝐶) = (Id‘𝐶)
5 ringcsect.n . . 3 𝑆 = (Sect‘𝐶)
6 ringcsect.u . . . 4 (𝜑𝑈𝑉)
7 ringcsect.c . . . . 5 𝐶 = (RingCat‘𝑈)
87ringccat 42349 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
96, 8syl 17 . . 3 (𝜑𝐶 ∈ Cat)
10 ringcsect.x . . 3 (𝜑𝑋𝐵)
11 ringcsect.y . . 3 (𝜑𝑌𝐵)
121, 2, 3, 4, 5, 9, 10, 11issect 16460 . 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, 11ringchom 42338 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RingHom 𝑌))
1413eleq2d 2716 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingHom 𝑌)))
157, 1, 6, 2, 11, 10ringchom 42338 . . . . . . 7 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RingHom 𝑋))
1615eleq2d 2716 . . . . . 6 (𝜑 → (𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ↔ 𝐺 ∈ (𝑌 RingHom 𝑋)))
1714, 16anbi12d 747 . . . . 5 (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))))
1817anbi1d 741 . . . 4 (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋))))
196adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑈𝑉)
2010adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑋𝐵)
217, 1, 6ringcbas 42336 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Ring))
2221eleq2d 2716 . . . . . . . . . 10 (𝜑 → (𝑋𝐵𝑋 ∈ (𝑈 ∩ Ring)))
23 inss1 3866 . . . . . . . . . . . 12 (𝑈 ∩ Ring) ⊆ 𝑈
2423a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑈 ∩ Ring) ⊆ 𝑈)
2524sseld 3635 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋𝑈))
2622, 25sylbid 230 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
2726adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → (𝑋𝐵𝑋𝑈))
2820, 27mpd 15 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑋𝑈)
2911adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑌𝐵)
3021eleq2d 2716 . . . . . . . . . 10 (𝜑 → (𝑌𝐵𝑌 ∈ (𝑈 ∩ Ring)))
3124sseld 3635 . . . . . . . . . 10 (𝜑 → (𝑌 ∈ (𝑈 ∩ Ring) → 𝑌𝑈))
3230, 31sylbid 230 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
3332adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → (𝑌𝐵𝑌𝑈))
3429, 33mpd 15 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑌𝑈)
35 eqid 2651 . . . . . . . . . 10 (Base‘𝑋) = (Base‘𝑋)
36 eqid 2651 . . . . . . . . . 10 (Base‘𝑌) = (Base‘𝑌)
3735, 36rhmf 18774 . . . . . . . . 9 (𝐹 ∈ (𝑋 RingHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
3837adantr 480 . . . . . . . 8 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
3938adantl 481 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
4036, 35rhmf 18774 . . . . . . . . 9 (𝐺 ∈ (𝑌 RingHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
4140adantl 481 . . . . . . . 8 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
4241adantl 481 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
437, 19, 3, 28, 34, 28, 39, 42ringcco 42342 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = (𝐺𝐹))
44 ringcsect.e . . . . . . . 8 𝐸 = (Base‘𝑋)
457, 1, 4, 6, 10, 44ringcid 42350 . . . . . . 7 (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
4645adantr 480 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
4743, 46eqeq12d 2666 . . . . 5 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋) ↔ (𝐺𝐹) = ( I ↾ 𝐸)))
4847pm5.32da 674 . . . 4 (𝜑 → (((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
4918, 48bitrd 268 . . 3 (𝜑 → (((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
50 df-3an 1056 . . 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 1056 . . 3 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸)) ↔ ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) ∧ (𝐺𝐹) = ( I ↾ 𝐸)))
5249, 50, 513bitr4g 303 . 2 (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐹) = ((Id‘𝐶)‘𝑋)) ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
5312, 52bitrd 268 1 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ∩ cin 3606   ⊆ wss 3607  ⟨cop 4216   class class class wbr 4685   I cid 5052   ↾ cres 5145   ∘ ccom 5147  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Sectcsect 16451  Ringcrg 18593   RingHom crh 18760  RingCatcringc 42328 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-fal 1529  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-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-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  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-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-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-hom 16013  df-cco 16014  df-0g 16149  df-cat 16376  df-cid 16377  df-homf 16378  df-sect 16454  df-ssc 16517  df-resc 16518  df-subc 16519  df-estrc 16810  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-grp 17472  df-ghm 17705  df-mgp 18536  df-ur 18548  df-ring 18595  df-rnghom 18763  df-ringc 42330 This theorem is referenced by:  ringcinv  42357
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