Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isnat2 | Structured version Visualization version GIF version |
Description: Property of being a natural transformation. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
natfval.1 | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
natfval.b | ⊢ 𝐵 = (Base‘𝐶) |
natfval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
natfval.j | ⊢ 𝐽 = (Hom ‘𝐷) |
natfval.o | ⊢ · = (comp‘𝐷) |
isnat2.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) |
isnat2.g | ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) |
Ref | Expression |
---|---|
isnat2 | ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉 · ((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝐺)𝑦)‘ℎ)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐺)‘𝑦))(𝐴‘𝑥))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfunc 17126 | . . . . 5 ⊢ Rel (𝐶 Func 𝐷) | |
2 | isnat2.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷)) | |
3 | 1st2nd 7732 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) | |
4 | 1, 2, 3 | sylancr 589 | . . . 4 ⊢ (𝜑 → 𝐹 = 〈(1st ‘𝐹), (2nd ‘𝐹)〉) |
5 | isnat2.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 Func 𝐷)) | |
6 | 1st2nd 7732 | . . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) | |
7 | 1, 5, 6 | sylancr 589 | . . . 4 ⊢ (𝜑 → 𝐺 = 〈(1st ‘𝐺), (2nd ‘𝐺)〉) |
8 | 4, 7 | oveq12d 7168 | . . 3 ⊢ (𝜑 → (𝐹𝑁𝐺) = (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉)) |
9 | 8 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ 𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉))) |
10 | natfval.1 | . . 3 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
11 | natfval.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
12 | natfval.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
13 | natfval.j | . . 3 ⊢ 𝐽 = (Hom ‘𝐷) | |
14 | natfval.o | . . 3 ⊢ · = (comp‘𝐷) | |
15 | 1st2ndbr 7735 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) | |
16 | 1, 2, 15 | sylancr 589 | . . 3 ⊢ (𝜑 → (1st ‘𝐹)(𝐶 Func 𝐷)(2nd ‘𝐹)) |
17 | 1st2ndbr 7735 | . . . 4 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐺 ∈ (𝐶 Func 𝐷)) → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺)) | |
18 | 1, 5, 17 | sylancr 589 | . . 3 ⊢ (𝜑 → (1st ‘𝐺)(𝐶 Func 𝐷)(2nd ‘𝐺)) |
19 | 10, 11, 12, 13, 14, 16, 18 | isnat 17211 | . 2 ⊢ (𝜑 → (𝐴 ∈ (〈(1st ‘𝐹), (2nd ‘𝐹)〉𝑁〈(1st ‘𝐺), (2nd ‘𝐺)〉) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉 · ((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝐺)𝑦)‘ℎ)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐺)‘𝑦))(𝐴‘𝑥))))) |
20 | 9, 19 | bitrd 281 | 1 ⊢ (𝜑 → (𝐴 ∈ (𝐹𝑁𝐺) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 (((1st ‘𝐹)‘𝑥)𝐽((1st ‘𝐺)‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀ℎ ∈ (𝑥𝐻𝑦)((𝐴‘𝑦)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐹)‘𝑦)〉 · ((1st ‘𝐺)‘𝑦))((𝑥(2nd ‘𝐹)𝑦)‘ℎ)) = (((𝑥(2nd ‘𝐺)𝑦)‘ℎ)(〈((1st ‘𝐹)‘𝑥), ((1st ‘𝐺)‘𝑥)〉 · ((1st ‘𝐺)‘𝑦))(𝐴‘𝑥))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 〈cop 4566 class class class wbr 5058 Rel wrel 5554 ‘cfv 6349 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 Xcixp 8455 Basecbs 16477 Hom chom 16570 compcco 16571 Func cfunc 17118 Nat cnat 17205 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-ixp 8456 df-func 17122 df-nat 17207 |
This theorem is referenced by: fuccocl 17228 fucidcl 17229 invfuc 17238 curf2cl 17475 yonedalem4c 17521 yonedalem3 17524 |
Copyright terms: Public domain | W3C validator |