Theorem inveq 17741

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	⊢ (𝜑 → 𝐶 ∈ Cat)
inveq.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
inveq.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
inveq	⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Proof of Theorem inveq

Step	Hyp	Ref	Expression
1		inveq.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2736	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
3		inveq.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat)
5		inveq.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌 ∈ 𝐵)
7		inveq.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑋 ∈ 𝐵)
9		inveq.n	. . . . . . . 8 ⊢ 𝑁 = (Inv‘𝐶)
10	1, 9, 3, 7, 5, 2	isinv 17727	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		simpr 484	. . . . . . 7 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
12	10, 11	biimtrdi 253	. . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
13	12	com12 32	. . . . 5 ⊢ (𝐹(𝑋𝑁𝑌)𝐺 → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
14	13	adantr 480	. . . 4 ⊢ ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
15	14	impcom 407	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
16	1, 9, 3, 7, 5, 2	isinv 17727	. . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹)))
17		simpl 482	. . . . . 6 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
18	16, 17	biimtrdi 253	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
19	18	adantld 490	. . . 4 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
20	19	imp 406	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
21	1, 2, 4, 6, 8, 15, 20	sectcan 17722	. 2 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾)
22	21	ex 412	1 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Catccat 17630  Sectcsect 17711  Invcinv 17712

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-cat 17634  df-cid 17635  df-sect 17714  df-inv 17715

This theorem is referenced by: (None)