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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 2735 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | natixp.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐷) | |
6 | eqid 2735 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | 2 | natrcl 18005 | . . . . . . 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 494 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
10 | df-br 5149 | . . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
11 | 9, 10 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
12 | 8 | simprd 495 | . . . . 5 ⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) |
13 | df-br 5149 | . . . . 5 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) | |
14 | 12, 13 | sylibr 234 | . . . 4 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) |
15 | 2, 3, 4, 5, 6, 11, 14 | isnat 18002 | . . 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 232 | . 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 494 | 1 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 〈cop 4637 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Xcixp 8936 Basecbs 17245 Hom chom 17309 compcco 17310 Func cfunc 17905 Nat cnat 17996 |
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-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-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-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-ixp 8937 df-func 17909 df-nat 17998 |
This theorem is referenced by: natcl 18008 natfn 18009 |
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