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Theorem inveq 17032
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 2818 . . 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 17018 . . . . . . 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 17018 . . . . . 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 17013 . 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 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  Catccat 16923  Sectcsect 17002  Invcinv 17003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-cat 16927  df-cid 16928  df-sect 17005  df-inv 17006
This theorem is referenced by: (None)
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