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Theorem inveq 17818
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 2737 . . 3 (Sect‘𝐶) = (Sect‘𝐶)
3 inveq.c . . . 4 (𝜑𝐶 ∈ Cat)
43adantr 480 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat)
5 inveq.y . . . 4 (𝜑𝑌𝐵)
65adantr 480 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌𝐵)
7 inveq.x . . . 4 (𝜑𝑋𝐵)
87adantr 480 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝑋𝐵)
9 inveq.n . . . . . . . 8 𝑁 = (Inv‘𝐶)
101, 9, 3, 7, 5, 2isinv 17804 . . . . . . 7 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11 simpr 484 . . . . . . 7 ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
1210, 11biimtrdi 253 . . . . . 6 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1312com12 32 . . . . 5 (𝐹(𝑋𝑁𝑌)𝐺 → (𝜑𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1413adantr 480 . . . 4 ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → (𝜑𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
1514impcom 407 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
161, 9, 3, 7, 5, 2isinv 17804 . . . . . 6 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾𝐾(𝑌(Sect‘𝐶)𝑋)𝐹)))
17 simpl 482 . . . . . 6 ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
1816, 17biimtrdi 253 . . . . 5 (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
1918adantld 490 . . . 4 (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
2019imp 406 . . 3 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
211, 2, 4, 6, 8, 15, 20sectcan 17799 . 2 ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾)
2221ex 412 1 (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  Catccat 17707  Sectcsect 17788  Invcinv 17789
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
This theorem is referenced by: (None)
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