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Theorem rcaninv 16277
 Description: Right cancellation of an inverse of an isomorphism. (Contributed by AV, 5-Apr-2017.)
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 2610 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2610 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 rcaninv.c . . . . . 6 (𝜑𝐶 ∈ Cat)
5 rcaninv.y . . . . . 6 (𝜑𝑌𝐵)
6 rcaninv.x . . . . . 6 (𝜑𝑋𝐵)
7 eqid 2610 . . . . . . . 8 (Iso‘𝐶) = (Iso‘𝐶)
81, 2, 7, 4, 5, 6isohom 16259 . . . . . . 7 (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9 rcaninv.f . . . . . . 7 (𝜑𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
108, 9sseldd 3569 . . . . . 6 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
111, 2, 7, 4, 6, 5isohom 16259 . . . . . . 7 (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
12 rcaninv.n . . . . . . . . 9 𝑁 = (Inv‘𝐶)
131, 12, 4, 5, 6, 7invf 16251 . . . . . . . 8 (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
1413, 9ffvelrnd 6268 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
1511, 14sseldd 3569 . . . . . 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 16170 . . . . 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 2610 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
20 eqid 2610 . . . . . . . 8 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
211, 7, 12, 4, 5, 6, 9, 19, 20invcoisoid 16275 . . . . . . 7 (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2221eqcomd 2616 . . . . . 6 (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
2322oveq2d 6565 . . . . 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 16168 . . . . 5 (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
2518, 23, 243eqtr2rd 2651 . . . 4 (𝜑𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
2625adantr 480 . . 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 2619 . . . . . . . 8 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) =
2928a1i 11 . . . . . . 7 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = )
30 eqidd 2611 . . . . . . 7 (𝜑𝐺 = 𝐺)
31 rcaninv.1 . . . . . . . . 9 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
3231eqcomi 2619 . . . . . . . 8 ((𝑌𝑁𝑋)‘𝐹) = 𝑅
3332a1i 11 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
3429, 30, 33oveq123d 6570 . . . . . 6 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 𝑅))
3534adantr 480 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 𝑅))
36 simpr 476 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺 𝑅) = (𝐻 𝑅))
3735, 36eqtrd 2644 . . . 4 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 𝑅))
3837oveq1d 6564 . . 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 6562 . . . . . . 7 (𝐻 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
4039oveq1i 6559 . . . . . 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 2692 . . . . . . 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 16170 . . . . . 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 6559 . . . . . . . 8 (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
4645oveq2i 6560 . . . . . . 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 6565 . . . . . 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 2648 . . . . 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 16168 . . . . 5 (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
5141, 49, 503eqtrd 2648 . . . 4 (𝜑 → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
5251adantr 480 . . 3 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
5326, 38, 523eqtrd 2648 . 2 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → 𝐺 = 𝐻)
5453ex 449 1 (𝜑 → ((𝐺 𝑅) = (𝐻 𝑅) → 𝐺 = 𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  Idccid 16149  Invcinv 16228  Isociso 16229 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231  df-iso 16232 This theorem is referenced by:  initoeu2lem0  16486
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