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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 2798 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | eqid 2798 | . . . 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 2798 | . . . . . . 7 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
9 | sectcan.s | . . . . . . 7 ⊢ 𝑆 = (Sect‘𝐶) | |
10 | 1, 2, 3, 8, 9, 4, 5, 6 | issect 17015 | . . . . . 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 235 | . . . . 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 1139 | . . . 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 17015 | . . . . . 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 235 | . . . . 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 1139 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) |
17 | 15 | simp2d 1140 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17 | catass 16949 | . . 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 1141 | . . . 4 ⊢ (𝜑 → (𝐻(〈𝑌, 𝑋〉(comp‘𝐶)𝑌)𝐹) = ((Id‘𝐶)‘𝑌)) |
20 | 19 | oveq1d 7150 | . . 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 1141 | . . . 4 ⊢ (𝜑 → (𝐹(〈𝑋, 𝑌〉(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)) |
22 | 21 | oveq2d 7151 | . . 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 2841 | . 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 16946 | . 2 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)𝐺) = 𝐺) |
25 | 1, 2, 8, 4, 5, 3, 6, 17 | catrid 16947 | . 2 ⊢ (𝜑 → (𝐻(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = 𝐻) |
26 | 23, 24, 25 | 3eqtr3d 2841 | 1 ⊢ (𝜑 → 𝐺 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 〈cop 4531 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 compcco 16569 Catccat 16927 Idccid 16928 Sectcsect 17006 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-cat 16931 df-cid 16932 df-sect 17009 |
This theorem is referenced by: invfun 17026 inveq 17036 |
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