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Theorem ringcsect 41802
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 2621 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2621 . . 3 (comp‘𝐶) = (comp‘𝐶)
4 eqid 2621 . . 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 41795 . . . 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 16407 . 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 41784 . . . . . . 7 (𝜑 → (𝑋(Hom ‘𝐶)𝑌) = (𝑋 RingHom 𝑌))
1413eleq2d 2686 . . . . . 6 (𝜑 → (𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌) ↔ 𝐹 ∈ (𝑋 RingHom 𝑌)))
157, 1, 6, 2, 11, 10ringchom 41784 . . . . . . 7 (𝜑 → (𝑌(Hom ‘𝐶)𝑋) = (𝑌 RingHom 𝑋))
1615eleq2d 2686 . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑈𝑉)
2010adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑋𝐵)
217, 1, 6ringcbas 41782 . . . . . . . . . . 11 (𝜑𝐵 = (𝑈 ∩ Ring))
2221eleq2d 2686 . . . . . . . . . 10 (𝜑 → (𝑋𝐵𝑋 ∈ (𝑈 ∩ Ring)))
23 inss1 3831 . . . . . . . . . . . 12 (𝑈 ∩ Ring) ⊆ 𝑈
2423a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑈 ∩ Ring) ⊆ 𝑈)
2524sseld 3600 . . . . . . . . . 10 (𝜑 → (𝑋 ∈ (𝑈 ∩ Ring) → 𝑋𝑈))
2622, 25sylbid 230 . . . . . . . . 9 (𝜑 → (𝑋𝐵𝑋𝑈))
2726adantr 481 . . . . . . . 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 481 . . . . . . . 8 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝑌𝐵)
3021eleq2d 2686 . . . . . . . . . 10 (𝜑 → (𝑌𝐵𝑌 ∈ (𝑈 ∩ Ring)))
3124sseld 3600 . . . . . . . . . 10 (𝜑 → (𝑌 ∈ (𝑈 ∩ Ring) → 𝑌𝑈))
3230, 31sylbid 230 . . . . . . . . 9 (𝜑 → (𝑌𝐵𝑌𝑈))
3332adantr 481 . . . . . . . 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 2621 . . . . . . . . . 10 (Base‘𝑋) = (Base‘𝑋)
36 eqid 2621 . . . . . . . . . 10 (Base‘𝑌) = (Base‘𝑌)
3735, 36rhmf 18720 . . . . . . . . 9 (𝐹 ∈ (𝑋 RingHom 𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
3837adantr 481 . . . . . . . 8 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
3938adantl 482 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))
4036, 35rhmf 18720 . . . . . . . . 9 (𝐺 ∈ (𝑌 RingHom 𝑋) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
4140adantl 482 . . . . . . . 8 ((𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋)) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
4241adantl 482 . . . . . . 7 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → 𝐺:(Base‘𝑌)⟶(Base‘𝑋))
437, 19, 3, 28, 34, 28, 39, 42ringcco 41788 . . . . . 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 41796 . . . . . . 7 (𝜑 → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
4645adantr 481 . . . . . 6 ((𝜑 ∧ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋))) → ((Id‘𝐶)‘𝑋) = ( I ↾ 𝐸))
4743, 46eqeq12d 2636 . . . . 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 673 . . . 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 1039 . . 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 1039 . . 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 384  w3a 1037   = wceq 1482  wcel 1989  cin 3571  wss 3572  cop 4181   class class class wbr 4651   I cid 5021  cres 5114  ccom 5116  wf 5882  cfv 5886  (class class class)co 6647  Basecbs 15851  Hom chom 15946  compcco 15947  Catccat 16319  Idccid 16320  Sectcsect 16398  Ringcrg 18541   RingHom crh 18706  RingCatcringc 41774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-fal 1488  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-map 7856  df-pm 7857  df-ixp 7906  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-nn 11018  df-2 11076  df-3 11077  df-4 11078  df-5 11079  df-6 11080  df-7 11081  df-8 11082  df-9 11083  df-n0 11290  df-z 11375  df-dec 11491  df-uz 11685  df-fz 12324  df-struct 15853  df-ndx 15854  df-slot 15855  df-base 15857  df-sets 15858  df-ress 15859  df-plusg 15948  df-hom 15960  df-cco 15961  df-0g 16096  df-cat 16323  df-cid 16324  df-homf 16325  df-sect 16401  df-ssc 16464  df-resc 16465  df-subc 16466  df-estrc 16757  df-mgm 17236  df-sgrp 17278  df-mnd 17289  df-mhm 17329  df-grp 17419  df-ghm 17652  df-mgp 18484  df-ur 18496  df-ring 18543  df-rnghom 18709  df-ringc 41776
This theorem is referenced by:  ringcinv  41803
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