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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincinv | Structured version Visualization version GIF version |
Description: In a thin category, 𝐹 is an inverse of 𝐺 iff 𝐹 is a section of 𝐺 (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
thincsect.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincsect.b | ⊢ 𝐵 = (Base‘𝐶) |
thincsect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
thincsect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
thincsect.s | ⊢ 𝑆 = (Sect‘𝐶) |
thincinv.n | ⊢ 𝑁 = (Inv‘𝐶) |
Ref | Expression |
---|---|
thincinv | ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thincsect.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
2 | thincinv.n | . . 3 ⊢ 𝑁 = (Inv‘𝐶) | |
3 | thincsect.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
4 | 3 | thinccd 48825 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) |
5 | thincsect.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | thincsect.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | thincsect.s | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
8 | 1, 2, 4, 5, 6, 7 | isinv 17808 | . 2 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋𝑆𝑌)𝐺 ∧ 𝐺(𝑌𝑆𝑋)𝐹))) |
9 | 3, 1, 5, 6, 7 | thincsect2 48859 | . . 3 ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ 𝐺(𝑌𝑆𝑋)𝐹)) |
10 | 9 | biimpa 476 | . 2 ⊢ ((𝜑 ∧ 𝐹(𝑋𝑆𝑌)𝐺) → 𝐺(𝑌𝑆𝑋)𝐹) |
11 | 8, 10 | mpbiran3d 48646 | 1 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ 𝐹(𝑋𝑆𝑌)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Sectcsect 17792 Invcinv 17793 ThinCatcthinc 48819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-cat 17713 df-cid 17714 df-sect 17795 df-inv 17796 df-thinc 48820 |
This theorem is referenced by: (None) |
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