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Theorem inveq 17725
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 𝐡 = (Baseβ€˜πΆ)
inveq.n 𝑁 = (Invβ€˜πΆ)
inveq.c (πœ‘ β†’ 𝐢 ∈ Cat)
inveq.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
inveq.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
inveq (πœ‘ β†’ ((𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾) β†’ 𝐺 = 𝐾))

Proof of Theorem inveq
StepHypRef Expression
1 inveq.b . . 3 𝐡 = (Baseβ€˜πΆ)
2 eqid 2732 . . 3 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
3 inveq.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
43adantr 481 . . 3 ((πœ‘ ∧ (𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾)) β†’ 𝐢 ∈ Cat)
5 inveq.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
65adantr 481 . . 3 ((πœ‘ ∧ (𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾)) β†’ π‘Œ ∈ 𝐡)
7 inveq.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
87adantr 481 . . 3 ((πœ‘ ∧ (𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾)) β†’ 𝑋 ∈ 𝐡)
9 inveq.n . . . . . . . 8 𝑁 = (Invβ€˜πΆ)
101, 9, 3, 7, 5, 2isinv 17711 . . . . . . 7 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 ↔ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)))
11 simpr 485 . . . . . . 7 ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐺 ∧ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) β†’ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
1210, 11syl6bi 252 . . . . . 6 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐺 β†’ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹))
1312com12 32 . . . . 5 (𝐹(π‘‹π‘π‘Œ)𝐺 β†’ (πœ‘ β†’ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹))
1413adantr 481 . . . 4 ((𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾) β†’ (πœ‘ β†’ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹))
1514impcom 408 . . 3 ((πœ‘ ∧ (𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾)) β†’ 𝐺(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)
161, 9, 3, 7, 5, 2isinv 17711 . . . . . 6 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐾 ↔ (𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐾 ∧ 𝐾(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹)))
17 simpl 483 . . . . . 6 ((𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐾 ∧ 𝐾(π‘Œ(Sectβ€˜πΆ)𝑋)𝐹) β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐾)
1816, 17syl6bi 252 . . . . 5 (πœ‘ β†’ (𝐹(π‘‹π‘π‘Œ)𝐾 β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐾))
1918adantld 491 . . . 4 (πœ‘ β†’ ((𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾) β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐾))
2019imp 407 . . 3 ((πœ‘ ∧ (𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾)) β†’ 𝐹(𝑋(Sectβ€˜πΆ)π‘Œ)𝐾)
211, 2, 4, 6, 8, 15, 20sectcan 17706 . 2 ((πœ‘ ∧ (𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾)) β†’ 𝐺 = 𝐾)
2221ex 413 1 (πœ‘ β†’ ((𝐹(π‘‹π‘π‘Œ)𝐺 ∧ 𝐹(π‘‹π‘π‘Œ)𝐾) β†’ 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  Catccat 17612  Sectcsect 17695  Invcinv 17696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-cat 17616  df-cid 17617  df-sect 17698  df-inv 17699
This theorem is referenced by: (None)
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