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Theorem rcaninv 16765
Description: Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2020.)
Hypotheses
Ref Expression
rcaninv.b 𝐵 = (Base‘𝐶)
rcaninv.n 𝑁 = (Inv‘𝐶)
rcaninv.c (𝜑𝐶 ∈ Cat)
rcaninv.x (𝜑𝑋𝐵)
rcaninv.y (𝜑𝑌𝐵)
rcaninv.z (𝜑𝑍𝐵)
rcaninv.f (𝜑𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
rcaninv.g (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
rcaninv.h (𝜑𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
rcaninv.1 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
rcaninv.o = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)
Assertion
Ref Expression
rcaninv (𝜑 → ((𝐺 𝑅) = (𝐻 𝑅) → 𝐺 = 𝐻))

Proof of Theorem rcaninv
StepHypRef Expression
1 rcaninv.b . . . . . 6 𝐵 = (Base‘𝐶)
2 eqid 2797 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2797 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 rcaninv.c . . . . . 6 (𝜑𝐶 ∈ Cat)
5 rcaninv.y . . . . . 6 (𝜑𝑌𝐵)
6 rcaninv.x . . . . . 6 (𝜑𝑋𝐵)
7 eqid 2797 . . . . . . . 8 (Iso‘𝐶) = (Iso‘𝐶)
81, 2, 7, 4, 5, 6isohom 16747 . . . . . . 7 (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9 rcaninv.f . . . . . . 7 (𝜑𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
108, 9sseldd 3797 . . . . . 6 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
111, 2, 7, 4, 6, 5isohom 16747 . . . . . . 7 (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
12 rcaninv.n . . . . . . . . 9 𝑁 = (Inv‘𝐶)
131, 12, 4, 5, 6, 7invf 16739 . . . . . . . 8 (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
1413, 9ffvelrnd 6584 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
1511, 14sseldd 3797 . . . . . 6 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌))
16 rcaninv.z . . . . . 6 (𝜑𝑍𝐵)
17 rcaninv.g . . . . . 6 (𝜑𝐺 ∈ (𝑌(Hom ‘𝐶)𝑍))
181, 2, 3, 4, 5, 6, 5, 10, 15, 16, 17catass 16658 . . . . 5 (𝜑 → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
19 eqid 2797 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
20 eqid 2797 . . . . . . . 8 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
211, 7, 12, 4, 5, 6, 9, 19, 20invcoisoid 16763 . . . . . . 7 (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2221eqcomd 2803 . . . . . 6 (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
2322oveq2d 6892 . . . . 5 (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
241, 2, 19, 4, 5, 3, 16, 17catrid 16656 . . . . 5 (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
2518, 23, 243eqtr2rd 2838 . . . 4 (𝜑𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
2625adantr 473 . . 3 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → 𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
27 rcaninv.o . . . . . . . . 9 = (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)
2827eqcomi 2806 . . . . . . . 8 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) =
2928a1i 11 . . . . . . 7 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = )
30 eqidd 2798 . . . . . . 7 (𝜑𝐺 = 𝐺)
31 rcaninv.1 . . . . . . . . 9 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
3231eqcomi 2806 . . . . . . . 8 ((𝑌𝑁𝑋)‘𝐹) = 𝑅
3332a1i 11 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
3429, 30, 33oveq123d 6897 . . . . . 6 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 𝑅))
3534adantr 473 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 𝑅))
36 simpr 478 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺 𝑅) = (𝐻 𝑅))
3735, 36eqtrd 2831 . . . 4 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 𝑅))
3837oveq1d 6891 . . 3 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
3927oveqi 6889 . . . . . . 7 (𝐻 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
4039oveq1i 6886 . . . . . 6 ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹)
4140a1i 11 . . . . 5 (𝜑 → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
4231, 15syl5eqel 2880 . . . . . . 7 (𝜑𝑅 ∈ (𝑋(Hom ‘𝐶)𝑌))
43 rcaninv.h . . . . . . 7 (𝜑𝐻 ∈ (𝑌(Hom ‘𝐶)𝑍))
441, 2, 3, 4, 5, 6, 5, 10, 42, 16, 43catass 16658 . . . . . 6 (𝜑 → ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
4531oveq1i 6886 . . . . . . . 8 (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
4645oveq2i 6887 . . . . . . 7 (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
4746a1i 11 . . . . . 6 (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)))
4821oveq2d 6892 . . . . . 6 (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)(((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)))
4944, 47, 483eqtrd 2835 . . . . 5 (𝜑 → ((𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)))
501, 2, 19, 4, 5, 3, 16, 43catrid 16656 . . . . 5 (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
5141, 49, 503eqtrd 2835 . . . 4 (𝜑 → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
5251adantr 473 . . 3 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
5326, 38, 523eqtrd 2835 . 2 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → 𝐺 = 𝐻)
5453ex 402 1 (𝜑 → ((𝐺 𝑅) = (𝐻 𝑅) → 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cop 4372  cfv 6099  (class class class)co 6876  Basecbs 16181  Hom chom 16275  compcco 16276  Catccat 16636  Idccid 16637  Invcinv 16716  Isociso 16717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-cat 16640  df-cid 16641  df-sect 16718  df-inv 16719  df-iso 16720
This theorem is referenced by:  initoeu2lem0  16974
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