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Mirrors > Home > MPE Home > Th. List > arwlid | Structured version Visualization version GIF version |
Description: Left identity of a category using arrow notation. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwlid.h | ⊢ 𝐻 = (Homa‘𝐶) |
arwlid.o | ⊢ · = (compa‘𝐶) |
arwlid.a | ⊢ 1 = (Ida‘𝐶) |
arwlid.f | ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) |
Ref | Expression |
---|---|
arwlid | ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwlid.a | . . . . . 6 ⊢ 1 = (Ida‘𝐶) | |
2 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | arwlid.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
4 | arwlid.h | . . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | 4 | homarcl 18082 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqid 2735 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
8 | 4, 2 | homarcl2 18089 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
9 | 3, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
10 | 9 | simprd 495 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
11 | 1, 2, 6, 7, 10 | ida2 18113 | . . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
12 | 11 | oveq1d 7446 | . . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹))) |
13 | eqid 2735 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
14 | 9 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
15 | eqid 2735 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
16 | 4, 13 | homahom 18093 | . . . . . 6 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌)) |
17 | 3, 16 | syl 17 | . . . . 5 ⊢ (𝜑 → (2nd ‘𝐹) ∈ (𝑋(Hom ‘𝐶)𝑌)) |
18 | 2, 13, 7, 6, 14, 15, 10, 17 | catlid 17728 | . . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹)) |
19 | 12, 18 | eqtrd 2775 | . . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹)) |
20 | 19 | oteq3d 4892 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹))〉 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
21 | arwlid.o | . . 3 ⊢ · = (compa‘𝐶) | |
22 | 1, 2, 6, 10, 4 | idahom 18114 | . . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌)) |
23 | 21, 4, 3, 22, 15 | coaval 18122 | . 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 〈𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹))〉) |
24 | 4 | homadmcd 18096 | . . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
25 | 3, 24 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
26 | 20, 23, 25 | 3eqtr4d 2785 | 1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 〈cotp 4639 ‘cfv 6563 (class class class)co 7431 2nd c2nd 8012 Basecbs 17245 Hom chom 17309 compcco 17310 Catccat 17709 Idccid 17710 Homachoma 18077 Idacida 18107 compaccoa 18108 |
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-ot 4640 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-doma 18078 df-coda 18079 df-homa 18080 df-arw 18081 df-ida 18109 df-coa 18110 |
This theorem is referenced by: (None) |
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