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Theorem inveq 16641
Description: If there are two inverses of an 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 2771 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
3 inveq.c . . . 4 (𝜑𝐶 ∈ Cat)
43adantr 466 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat)
5 inveq.y . . . 4 (𝜑𝑌𝐵)
65adantr 466 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌𝐵)
7 inveq.x . . . 4 (𝜑𝑋𝐵)
87adantr 466 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝑋𝐵)
9 inveq.n . . . . . . . 8 𝑁 = (Inv‘𝐶)
101, 9, 3, 7, 5, 2isinv 16627 . . . . . . 7 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 simpr 471 . . . . . . 7 ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
1210, 11syl6bi 243 . . . . . 6 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1312com12 32 . . . . 5 (𝐹(𝑋𝑁𝑌)𝐺 → (𝜑𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1413adantr 466 . . . 4 ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → (𝜑𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1514impcom 394 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
161, 9, 3, 7, 5, 2isinv 16627 . . . . . 6 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾𝐾(𝑌(Sect‘𝐶)𝑋)𝐹)))
17 simpl 468 . . . . . 6 ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
1816, 17syl6bi 243 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
1918adantld 478 . . . 4 (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
2019imp 393 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
211, 2, 4, 6, 8, 15, 20sectcan 16622 . 2 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾)
2221ex 397 1 (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  Basecbs 16064  Catccat 16532  Sectcsect 16611  Invcinv 16612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-cat 16536  df-cid 16537  df-sect 16614  df-inv 16615
This theorem is referenced by: (None)
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