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Mirrors > Home > MPE Home > Th. List > inveq | Structured version Visualization version GIF version |
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.) |
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) |
Ref | Expression |
---|---|
inveq | ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inveq.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2733 | . . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
3 | inveq.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | 3 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat) |
5 | inveq.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 5 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌 ∈ 𝐵) |
7 | inveq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 7 | adantr 482 | . . 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 17702 | . . . . . . 7 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))) |
11 | simpr 486 | . . . . . . 7 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) | |
12 | 10, 11 | syl6bi 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 482 | . . . 4 ⊢ ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)) |
15 | 14 | impcom 409 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) |
16 | 1, 9, 3, 7, 5, 2 | isinv 17702 | . . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹))) |
17 | simpl 484 | . . . . . 6 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾) | |
18 | 16, 17 | syl6bi 253 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)) |
19 | 18 | adantld 492 | . . . 4 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)) |
20 | 19 | imp 408 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾) |
21 | 1, 2, 4, 6, 8, 15, 20 | sectcan 17697 | . 2 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾) |
22 | 21 | ex 414 | 1 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5146 ‘cfv 6539 (class class class)co 7403 Basecbs 17139 Catccat 17603 Sectcsect 17686 Invcinv 17687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-iun 4997 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-1st 7969 df-2nd 7970 df-cat 17607 df-cid 17608 df-sect 17689 df-inv 17690 |
This theorem is referenced by: (None) |
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