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Mirrors > Home > MPE Home > Th. List > arwrid | Structured version Visualization version GIF version |
Description: Right 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 |
---|---|
arwrid | ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwlid.a | . . . . . 6 ⊢ 1 = (Ida‘𝐶) | |
2 | eqid 2739 | . . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | arwlid.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌)) | |
4 | arwlid.h | . . . . . . . 8 ⊢ 𝐻 = (Homa‘𝐶) | |
5 | 4 | homarcl 17412 | . . . . . . 7 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat) |
6 | 3, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) |
7 | eqid 2739 | . . . . . 6 ⊢ (Id‘𝐶) = (Id‘𝐶) | |
8 | 4, 2 | homarcl2 17419 | . . . . . . . 8 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
9 | 3, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶))) |
10 | 9 | simpld 498 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
11 | 1, 2, 6, 7, 10 | ida2 17443 | . . . . 5 ⊢ (𝜑 → (2nd ‘( 1 ‘𝑋)) = ((Id‘𝐶)‘𝑋)) |
12 | 11 | oveq2d 7198 | . . . 4 ⊢ (𝜑 → ((2nd ‘𝐹)(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋))) = ((2nd ‘𝐹)(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋))) |
13 | eqid 2739 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
14 | eqid 2739 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
15 | 9 | simprd 499 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
16 | 4, 13 | homahom 17423 | . . . . . 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, 10, 14, 15, 17 | catrid 17070 | . . . 4 ⊢ (𝜑 → ((2nd ‘𝐹)(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)((Id‘𝐶)‘𝑋)) = (2nd ‘𝐹)) |
19 | 12, 18 | eqtrd 2774 | . . 3 ⊢ (𝜑 → ((2nd ‘𝐹)(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋))) = (2nd ‘𝐹)) |
20 | 19 | oteq3d 4785 | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌, ((2nd ‘𝐹)(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋)))〉 = 〈𝑋, 𝑌, (2nd ‘𝐹)〉) |
21 | arwlid.o | . . 3 ⊢ · = (compa‘𝐶) | |
22 | 1, 2, 6, 10, 4 | idahom 17444 | . . 3 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐻𝑋)) |
23 | 21, 4, 22, 3, 14 | coaval 17452 | . 2 ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 〈𝑋, 𝑌, ((2nd ‘𝐹)(〈𝑋, 𝑋〉(comp‘𝐶)𝑌)(2nd ‘( 1 ‘𝑋)))〉) |
24 | 4 | homadmcd 17426 | . . 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 2784 | 1 ⊢ (𝜑 → (𝐹 · ( 1 ‘𝑋)) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 〈cop 4532 〈cotp 4534 ‘cfv 6349 (class class class)co 7182 2nd c2nd 7725 Basecbs 16598 Hom chom 16691 compcco 16692 Catccat 17050 Idccid 17051 Homachoma 17407 Idacida 17437 compaccoa 17438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-ot 4535 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-1st 7726 df-2nd 7727 df-cat 17054 df-cid 17055 df-doma 17408 df-coda 17409 df-homa 17410 df-arw 17411 df-ida 17439 df-coa 17440 |
This theorem is referenced by: (None) |
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