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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 2736 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | arwlid.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
4 | arwlid.h | . . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | 4 | homarcl 17488 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqid 2736 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
8 | 4, 2 | homarcl2 17495 | . . . . . . . 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 499 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
11 | 1, 2, 6, 7, 10 | ida2 17519 | . . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑌)) = ((Id‘𝐶)‘𝑌)) |
12 | 11 | oveq1d 7206 | . . . 4 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹))) |
13 | eqid 2736 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
14 | 9 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
15 | eqid 2736 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
16 | 4, 13 | homahom 17499 | . . . . . 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 17140 | . . . 4 ⊢ (𝜑 → (((Id‘𝐶)‘𝑌)(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹)) |
19 | 12, 18 | eqtrd 2771 | . . 3 ⊢ (𝜑 → ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹)) = (2nd ‘𝐹)) |
20 | 19 | oteq3d 4784 | . 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 17520 | . . 3 ⊢ (𝜑 → ( 1 ‘𝑌) ∈ (𝑌𝐻𝑌)) |
23 | 21, 4, 3, 22, 15 | coaval 17528 | . 2 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 〈𝑋, 𝑌, ((2nd ‘( 1 ‘𝑌))(〈𝑋, 𝑌〉(comp‘𝐶)𝑌)(2nd ‘𝐹))〉) |
24 | 4 | homadmcd 17502 | . . 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 2781 | 1 ⊢ (𝜑 → (( 1 ‘𝑌) · 𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 〈cop 4533 〈cotp 4535 ‘cfv 6358 (class class class)co 7191 2nd c2nd 7738 Basecbs 16666 Hom chom 16760 compcco 16761 Catccat 17121 Idccid 17122 Homachoma 17483 Idacida 17513 compaccoa 17514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-ot 4536 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-cat 17125 df-cid 17126 df-doma 17484 df-coda 17485 df-homa 17486 df-arw 17487 df-ida 17515 df-coa 17516 |
This theorem is referenced by: (None) |
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