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Mirrors > Home > MPE Home > Th. List > natixp | Structured version Visualization version GIF version |
Description: A natural transformation is a function from the objects of 𝐶 to homomorphisms from 𝐹(𝑥) to 𝐺(𝑥). (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
natrcl.1 | ⊢ 𝑁 = (𝐶 Nat 𝐷) |
natixp.2 | ⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉)) |
natixp.b | ⊢ 𝐵 = (Base‘𝐶) |
natixp.j | ⊢ 𝐽 = (Hom ‘𝐷) |
Ref | Expression |
---|---|
natixp | ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | natixp.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉)) | |
2 | natrcl.1 | . . . 4 ⊢ 𝑁 = (𝐶 Nat 𝐷) | |
3 | natixp.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
4 | eqid 2793 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | natixp.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐷) | |
6 | eqid 2793 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | 2 | natrcl 17037 | . . . . . . 7 ⊢ (𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉) → (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) ∧ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷))) |
8 | 1, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷) ∧ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷))) |
9 | 8 | simpld 495 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
10 | df-br 4957 | . . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
11 | 9, 10 | sylibr 235 | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
12 | 8 | simprd 496 | . . . . 5 ⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) |
13 | df-br 4957 | . . . . 5 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) | |
14 | 12, 13 | sylibr 235 | . . . 4 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) |
15 | 2, 3, 4, 5, 6, 11, 14 | isnat 17034 | . . 3 ⊢ (𝜑 → (𝐴 ∈ (〈𝐹, 𝐺〉𝑁〈𝐾, 𝐿〉) ↔ (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐷)(𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉(comp‘𝐷)(𝐾‘𝑦))(𝐴‘𝑥))))) |
16 | 1, 15 | mpbid 233 | . 2 ⊢ (𝜑 → (𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥)) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ (𝑥(Hom ‘𝐶)𝑦)((𝐴‘𝑦)(〈(𝐹‘𝑥), (𝐹‘𝑦)〉(comp‘𝐷)(𝐾‘𝑦))((𝑥𝐺𝑦)‘𝑧)) = (((𝑥𝐿𝑦)‘𝑧)(〈(𝐹‘𝑥), (𝐾‘𝑥)〉(comp‘𝐷)(𝐾‘𝑦))(𝐴‘𝑥)))) |
17 | 16 | simpld 495 | 1 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∀wral 3103 〈cop 4472 class class class wbr 4956 ‘cfv 6217 (class class class)co 7007 Xcixp 8300 Basecbs 16300 Hom chom 16393 compcco 16394 Func cfunc 16941 Nat cnat 17028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-ov 7010 df-oprab 7011 df-mpo 7012 df-1st 7536 df-2nd 7537 df-ixp 8301 df-func 16945 df-nat 17030 |
This theorem is referenced by: natcl 17040 natfn 17041 |
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