![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sectcan | Structured version Visualization version GIF version |
Description: If 𝐺 is a section of 𝐹 and 𝐹 is a section of 𝐻, then 𝐺 = 𝐻. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
sectcan.b | ⊢ 𝐵 = (Base‘𝐶) |
sectcan.s | ⊢ 𝑆 = (Sect‘𝐶) |
sectcan.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
sectcan.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
sectcan.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
sectcan.1 | ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹) |
sectcan.2 | ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻) |
Ref | Expression |
---|---|
sectcan | ⊢ (𝜑 → 𝐺 = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sectcan.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
2 | eqid 2735 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2735 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
4 | sectcan.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
5 | sectcan.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | sectcan.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | sectcan.1 | . . . . . 6 ⊢ (𝜑 → 𝐺(𝑋𝑆𝑌)𝐹) | |
8 | eqid 2735 | . . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | sectcan.s | . . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶) | |
10 | 1, 2, 3, 8, 9, 4, 5, 6 | issect 17801 | . . . . . 6 ⊢ (𝜑 → (𝐺(𝑋𝑆𝑌)𝐹 ↔ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)))) |
11 | 7, 10 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))) |
12 | 11 | simp1d 1141 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
13 | sectcan.2 | . . . . . 6 ⊢ (𝜑 → 𝐹(𝑌𝑆𝑋)𝐻) | |
14 | 1, 2, 3, 8, 9, 4, 6, 5 | issect 17801 | . . . . . 6 ⊢ (𝜑 → (𝐹(𝑌𝑆𝑋)𝐻 ↔ (𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)))) |
15 | 13, 14 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌))) |
16 | 15 | simp1d 1141 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) |
17 | 15 | simp2d 1142 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17 | catass 17731 | . . 3 ⊢ (𝜑 → ((𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺))) |
19 | 15 | simp3d 1143 | . . . 4 ⊢ (𝜑 → (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)) |
20 | 19 | oveq1d 7446 | . . 3 ⊢ (𝜑 → ((𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺)) |
21 | 11 | simp3d 1143 | . . . 4 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)) |
22 | 21 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺)) = (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋))) |
23 | 18, 20, 22 | 3eqtr3d 2783 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋))) |
24 | 1, 2, 8, 4, 5, 3, 6, 12 | catlid 17728 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = 𝐺) |
25 | 1, 2, 8, 4, 5, 3, 6, 17 | catrid 17729 | . 2 ⊢ (𝜑 → (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻) |
26 | 23, 24, 25 | 3eqtr3d 2783 | 1 ⊢ (𝜑 → 𝐺 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 compcco 17310 Catccat 17709 Idccid 17710 Sectcsect 17792 |
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 |
This theorem is referenced by: invfun 17812 inveq 17822 |
Copyright terms: Public domain | W3C validator |