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Theorem rcaninv 16435
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 2620 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2620 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 rcaninv.c . . . . . 6 (𝜑𝐶 ∈ Cat)
5 rcaninv.y . . . . . 6 (𝜑𝑌𝐵)
6 rcaninv.x . . . . . 6 (𝜑𝑋𝐵)
7 eqid 2620 . . . . . . . 8 (Iso‘𝐶) = (Iso‘𝐶)
81, 2, 7, 4, 5, 6isohom 16417 . . . . . . 7 (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9 rcaninv.f . . . . . . 7 (𝜑𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
108, 9sseldd 3596 . . . . . 6 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
111, 2, 7, 4, 6, 5isohom 16417 . . . . . . 7 (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
12 rcaninv.n . . . . . . . . 9 𝑁 = (Inv‘𝐶)
131, 12, 4, 5, 6, 7invf 16409 . . . . . . . 8 (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
1413, 9ffvelrnd 6346 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
1511, 14sseldd 3596 . . . . . 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 16328 . . . . 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 2620 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
20 eqid 2620 . . . . . . . 8 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
211, 7, 12, 4, 5, 6, 9, 19, 20invcoisoid 16433 . . . . . . 7 (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2221eqcomd 2626 . . . . . 6 (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
2322oveq2d 6651 . . . . 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 16326 . . . . 5 (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
2518, 23, 243eqtr2rd 2661 . . . 4 (𝜑𝐺 = ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹))(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹))
2625adantr 481 . . 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 2629 . . . . . . . 8 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) =
2928a1i 11 . . . . . . 7 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = )
30 eqidd 2621 . . . . . . 7 (𝜑𝐺 = 𝐺)
31 rcaninv.1 . . . . . . . . 9 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
3231eqcomi 2629 . . . . . . . 8 ((𝑌𝑁𝑋)‘𝐹) = 𝑅
3332a1i 11 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
3429, 30, 33oveq123d 6656 . . . . . 6 (𝜑 → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 𝑅))
3534adantr 481 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐺 𝑅))
36 simpr 477 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺 𝑅) = (𝐻 𝑅))
3735, 36eqtrd 2654 . . . 4 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 𝑅))
3837oveq1d 6650 . . 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 6648 . . . . . . 7 (𝐻 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
4039oveq1i 6645 . . . . . 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 2703 . . . . . . 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 16328 . . . . . 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 6645 . . . . . . . 8 (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
4645oveq2i 6646 . . . . . . 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 6651 . . . . . 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 2658 . . . . 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 16326 . . . . 5 (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
5141, 49, 503eqtrd 2658 . . . 4 (𝜑 → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
5251adantr 481 . . 3 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → ((𝐻 𝑅)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑍)𝐹) = 𝐻)
5326, 38, 523eqtrd 2658 . 2 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → 𝐺 = 𝐻)
5453ex 450 1 (𝜑 → ((𝐺 𝑅) = (𝐻 𝑅) → 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  cop 4174  cfv 5876  (class class class)co 6635  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Idccid 16307  Invcinv 16386  Isociso 16387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-cat 16310  df-cid 16311  df-sect 16388  df-inv 16389  df-iso 16390
This theorem is referenced by:  initoeu2lem0  16644
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