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Theorem inveq 17036
Description: If there are two inverses of a morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.) (Revised by AV, 3-Jul-2022.)
Hypotheses
Ref Expression
inveq.b 𝐵 = (Base‘𝐶)
inveq.n 𝑁 = (Inv‘𝐶)
inveq.c (𝜑𝐶 ∈ Cat)
inveq.x (𝜑𝑋𝐵)
inveq.y (𝜑𝑌𝐵)
Assertion
Ref Expression
inveq (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Proof of Theorem inveq
StepHypRef Expression
1 inveq.b . . 3 𝐵 = (Base‘𝐶)
2 eqid 2825 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
3 inveq.c . . . 4 (𝜑𝐶 ∈ Cat)
43adantr 481 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat)
5 inveq.y . . . 4 (𝜑𝑌𝐵)
65adantr 481 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌𝐵)
7 inveq.x . . . 4 (𝜑𝑋𝐵)
87adantr 481 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝑋𝐵)
9 inveq.n . . . . . . . 8 𝑁 = (Inv‘𝐶)
101, 9, 3, 7, 5, 2isinv 17022 . . . . . . 7 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 simpr 485 . . . . . . 7 ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
1210, 11syl6bi 254 . . . . . 6 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1312com12 32 . . . . 5 (𝐹(𝑋𝑁𝑌)𝐺 → (𝜑𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1413adantr 481 . . . 4 ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → (𝜑𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1514impcom 408 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
161, 9, 3, 7, 5, 2isinv 17022 . . . . . 6 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾𝐾(𝑌(Sect‘𝐶)𝑋)𝐹)))
17 simpl 483 . . . . . 6 ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
1816, 17syl6bi 254 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
1918adantld 491 . . . 4 (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
2019imp 407 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
211, 2, 4, 6, 8, 15, 20sectcan 17017 . 2 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾)
2221ex 413 1 (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107   class class class wbr 5062  cfv 6351  (class class class)co 7151  Basecbs 16475  Catccat 16927  Sectcsect 17006  Invcinv 17007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-cat 16931  df-cid 16932  df-sect 17009  df-inv 17010
This theorem is referenced by: (None)
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