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Theorem rcaninv 17838
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 2737 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
3 eqid 2737 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
4 rcaninv.c . . . . . 6 (𝜑𝐶 ∈ Cat)
5 rcaninv.y . . . . . 6 (𝜑𝑌𝐵)
6 rcaninv.x . . . . . 6 (𝜑𝑋𝐵)
7 eqid 2737 . . . . . . . 8 (Iso‘𝐶) = (Iso‘𝐶)
81, 2, 7, 4, 5, 6isohom 17820 . . . . . . 7 (𝜑 → (𝑌(Iso‘𝐶)𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
9 rcaninv.f . . . . . . 7 (𝜑𝐹 ∈ (𝑌(Iso‘𝐶)𝑋))
108, 9sseldd 3984 . . . . . 6 (𝜑𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
111, 2, 7, 4, 6, 5isohom 17820 . . . . . . 7 (𝜑 → (𝑋(Iso‘𝐶)𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
12 rcaninv.n . . . . . . . . 9 𝑁 = (Inv‘𝐶)
131, 12, 4, 5, 6, 7invf 17812 . . . . . . . 8 (𝜑 → (𝑌𝑁𝑋):(𝑌(Iso‘𝐶)𝑋)⟶(𝑋(Iso‘𝐶)𝑌))
1413, 9ffvelcdmd 7105 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) ∈ (𝑋(Iso‘𝐶)𝑌))
1511, 14sseldd 3984 . . . . . 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 17729 . . . . 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 2737 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
20 eqid 2737 . . . . . . . 8 (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌) = (⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)
211, 7, 12, 4, 5, 6, 9, 19, 20invcoisoid 17836 . . . . . . 7 (𝜑 → (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))
2221eqcomd 2743 . . . . . 6 (𝜑 → ((Id‘𝐶)‘𝑌) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹))
2322oveq2d 7447 . . . . 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 17727 . . . . 5 (𝜑 → (𝐺(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐺)
2518, 23, 243eqtr2rd 2784 . . . 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 2746 . . . . . . . 8 (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) =
2928a1i 11 . . . . . . 7 (𝜑 → (⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍) = )
30 eqidd 2738 . . . . . . 7 (𝜑𝐺 = 𝐺)
31 rcaninv.1 . . . . . . . . 9 𝑅 = ((𝑌𝑁𝑋)‘𝐹)
3231eqcomi 2746 . . . . . . . 8 ((𝑌𝑁𝑋)‘𝐹) = 𝑅
3332a1i 11 . . . . . . 7 (𝜑 → ((𝑌𝑁𝑋)‘𝐹) = 𝑅)
3429, 30, 33oveq123d 7452 . . . . . 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 484 . . . . 5 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺 𝑅) = (𝐻 𝑅))
3735, 36eqtrd 2777 . . . 4 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)((𝑌𝑁𝑋)‘𝐹)) = (𝐻 𝑅))
3837oveq1d 7446 . . 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 7444 . . . . . . 7 (𝐻 𝑅) = (𝐻(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑅)
4039oveq1i 7441 . . . . . 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, 15eqeltrid 2845 . . . . . . 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 17729 . . . . . 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 7441 . . . . . . . 8 (𝑅(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹) = (((𝑌𝑁𝑋)‘𝐹)(⟨𝑌, 𝑋⟩(comp‘𝐶)𝑌)𝐹)
4645oveq2i 7442 . . . . . . 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 7447 . . . . . 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 2781 . . . . 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 17727 . . . . 5 (𝜑 → (𝐻(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑍)((Id‘𝐶)‘𝑌)) = 𝐻)
5141, 49, 503eqtrd 2781 . . . 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 2781 . 2 ((𝜑 ∧ (𝐺 𝑅) = (𝐻 𝑅)) → 𝐺 = 𝐻)
5453ex 412 1 (𝜑 → ((𝐺 𝑅) = (𝐻 𝑅) → 𝐺 = 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cop 4632  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Idccid 17708  Invcinv 17789  Isociso 17790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-cat 17711  df-cid 17712  df-sect 17791  df-inv 17792  df-iso 17793
This theorem is referenced by:  initoeu2lem0  18058
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