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Mirrors > Home > MPE Home > Th. List > caofdir | Structured version Visualization version GIF version |
Description: Transfer a reverse distributive law to the function operation. (Contributed by NM, 19-Oct-2014.) |
Ref | Expression |
---|---|
caofdi.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
caofdi.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) |
caofdi.3 | ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) |
caofdi.4 | ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) |
caofdir.5 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧))) |
Ref | Expression |
---|---|
caofdir | ⊢ (𝜑 → ((𝐺 ∘𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺 ∘𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻 ∘𝑓 𝑇𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofdir.5 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧))) | |
2 | 1 | adantlr 751 | . . . 4 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝐾)) → ((𝑥𝑅𝑦)𝑇𝑧) = ((𝑥𝑇𝑧)𝑂(𝑦𝑇𝑧))) |
3 | caofdi.3 | . . . . 5 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) | |
4 | 3 | ffvelrnda 6359 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆) |
5 | caofdi.4 | . . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) | |
6 | 5 | ffvelrnda 6359 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆) |
7 | caofdi.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) | |
8 | 7 | ffvelrnda 6359 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾) |
9 | 2, 4, 6, 8 | caovdird 6852 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤)) = (((𝐺‘𝑤)𝑇(𝐹‘𝑤))𝑂((𝐻‘𝑤)𝑇(𝐹‘𝑤)))) |
10 | 9 | mpteq2dva 4744 | . 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤))) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑇(𝐹‘𝑤))𝑂((𝐻‘𝑤)𝑇(𝐹‘𝑤))))) |
11 | caofdi.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | ovexd 6680 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V) | |
13 | 3 | feqmptd 6249 | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
14 | 5 | feqmptd 6249 | . . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤))) |
15 | 11, 4, 6, 13, 14 | offval2 6914 | . . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤)))) |
16 | 7 | feqmptd 6249 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤))) |
17 | 11, 12, 8, 15, 16 | offval2 6914 | . 2 ⊢ (𝜑 → ((𝐺 ∘𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤)))) |
18 | ovexd 6680 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑇(𝐹‘𝑤)) ∈ V) | |
19 | ovexd 6680 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤)𝑇(𝐹‘𝑤)) ∈ V) | |
20 | 11, 4, 8, 13, 16 | offval2 6914 | . . 3 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑇(𝐹‘𝑤)))) |
21 | 11, 6, 8, 14, 16 | offval2 6914 | . . 3 ⊢ (𝜑 → (𝐻 ∘𝑓 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐻‘𝑤)𝑇(𝐹‘𝑤)))) |
22 | 11, 18, 19, 20, 21 | offval2 6914 | . 2 ⊢ (𝜑 → ((𝐺 ∘𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻 ∘𝑓 𝑇𝐹)) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑇(𝐹‘𝑤))𝑂((𝐻‘𝑤)𝑇(𝐹‘𝑤))))) |
23 | 10, 17, 22 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → ((𝐺 ∘𝑓 𝑅𝐻) ∘𝑓 𝑇𝐹) = ((𝐺 ∘𝑓 𝑇𝐹) ∘𝑓 𝑂(𝐻 ∘𝑓 𝑇𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ↦ cmpt 4729 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 |
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 |
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-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-of 6897 |
This theorem is referenced by: psrlmod 19401 lflvsdi1 34365 mendlmod 37763 expgrowth 38534 |
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