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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 2622 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
5 | natixp.j | . . . 4 ⊢ 𝐽 = (Hom ‘𝐷) | |
6 | eqid 2622 | . . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷) | |
7 | 2 | natrcl 16610 | . . . . . . 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 475 | . . . . 5 ⊢ (𝜑 → 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) |
10 | df-br 4654 | . . . . 5 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ 〈𝐹, 𝐺〉 ∈ (𝐶 Func 𝐷)) | |
11 | 9, 10 | sylibr 224 | . . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺) |
12 | 8 | simprd 479 | . . . . 5 ⊢ (𝜑 → 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) |
13 | df-br 4654 | . . . . 5 ⊢ (𝐾(𝐶 Func 𝐷)𝐿 ↔ 〈𝐾, 𝐿〉 ∈ (𝐶 Func 𝐷)) | |
14 | 12, 13 | sylibr 224 | . . . 4 ⊢ (𝜑 → 𝐾(𝐶 Func 𝐷)𝐿) |
15 | 2, 3, 4, 5, 6, 11, 14 | isnat 16607 | . . 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 222 | . 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 475 | 1 ⊢ (𝜑 → 𝐴 ∈ X𝑥 ∈ 𝐵 ((𝐹‘𝑥)𝐽(𝐾‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Xcixp 7908 Basecbs 15857 Hom chom 15952 compcco 15953 Func cfunc 16514 Nat cnat 16601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-ixp 7909 df-func 16518 df-nat 16603 |
This theorem is referenced by: natcl 16613 natfn 16614 |
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