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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 714 | . . . 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 | ffvelcdmda 7040 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆) |
5 | caofdi.4 | . . . . 5 ⊢ (𝜑 → 𝐻:𝐴⟶𝑆) | |
6 | 5 | ffvelcdmda 7040 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐻‘𝑤) ∈ 𝑆) |
7 | caofdi.2 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝐾) | |
8 | 7 | ffvelcdmda 7040 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐾) |
9 | 2, 4, 6, 8 | caovdird 7577 | . . 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 5210 | . 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 7397 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑅(𝐻‘𝑤)) ∈ V) | |
13 | 3 | feqmptd 6915 | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
14 | 5 | feqmptd 6915 | . . . 4 ⊢ (𝜑 → 𝐻 = (𝑤 ∈ 𝐴 ↦ (𝐻‘𝑤))) |
15 | 11, 4, 6, 13, 14 | offval2 7642 | . . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐻) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐻‘𝑤)))) |
16 | 7 | feqmptd 6915 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤))) |
17 | 11, 12, 8, 15, 16 | offval2 7642 | . 2 ⊢ (𝜑 → ((𝐺 ∘f 𝑅𝐻) ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ (((𝐺‘𝑤)𝑅(𝐻‘𝑤))𝑇(𝐹‘𝑤)))) |
18 | ovexd 7397 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐺‘𝑤)𝑇(𝐹‘𝑤)) ∈ V) | |
19 | ovexd 7397 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐻‘𝑤)𝑇(𝐹‘𝑤)) ∈ V) | |
20 | 11, 4, 8, 13, 16 | offval2 7642 | . . 3 ⊢ (𝜑 → (𝐺 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑇(𝐹‘𝑤)))) |
21 | 11, 6, 8, 14, 16 | offval2 7642 | . . 3 ⊢ (𝜑 → (𝐻 ∘f 𝑇𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐻‘𝑤)𝑇(𝐹‘𝑤)))) |
22 | 11, 18, 19, 20, 21 | offval2 7642 | . 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 2787 | 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ↦ cmpt 5193 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 ∘f cof 7620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 |
This theorem is referenced by: psrlmod 21386 lflvsdi1 37569 mendlmod 41549 expgrowth 42689 |
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