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Mirrors > Home > MPE Home > Th. List > inveq | Structured version Visualization version GIF version |
Description: If there are two inverses of an morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2017.) |
Ref | Expression |
---|---|
inveq.b | ⊢ 𝐵 = (Base‘𝐶) |
inveq.h | ⊢ 𝐻 = (Hom ‘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 2622 | . . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶) | |
3 | inveq.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
4 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat) |
5 | inveq.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌 ∈ 𝐵) |
7 | inveq.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | 7 | adantr 481 | . . 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 16420 | . . . . . . 7 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))) |
11 | simpr 477 | . . . . . . 7 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) | |
12 | 10, 11 | syl6bi 243 | . . . . . 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 481 | . . . 4 ⊢ ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)) |
15 | 14 | impcom 446 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) |
16 | 1, 9, 3, 7, 5, 2 | isinv 16420 | . . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹))) |
17 | simpl 473 | . . . . . 6 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾) | |
18 | 16, 17 | syl6bi 243 | . . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)) |
19 | 18 | adantld 483 | . . . 4 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)) |
20 | 19 | imp 445 | . . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾) |
21 | 1, 2, 4, 6, 8, 15, 20 | sectcan 16415 | . 2 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾) |
22 | 21 | ex 450 | 1 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Hom chom 15952 Catccat 16325 Sectcsect 16404 Invcinv 16405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-cat 16329 df-cid 16330 df-sect 16407 df-inv 16408 |
This theorem is referenced by: (None) |
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