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Mirrors > Home > MPE Home > Th. List > caofcom | Structured version Visualization version GIF version |
Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
caofref.2 | ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) |
caofcom.3 | ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) |
caofcom.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) |
Ref | Expression |
---|---|
caofcom | ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.2 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) | |
2 | 1 | ffvelrnda 6267 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
3 | caofcom.3 | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) | |
4 | 3 | ffvelrnda 6267 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆) |
5 | 2, 4 | jca 553 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) |
6 | caofcom.4 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) | |
7 | 6 | caovcomg 6727 | . . . 4 ⊢ ((𝜑 ∧ ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) |
8 | 5, 7 | syldan 486 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) |
9 | 8 | mpteq2dva 4672 | . 2 ⊢ (𝜑 → (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤))) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤)))) |
10 | caofref.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
11 | 1 | feqmptd 6159 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤))) |
12 | 3 | feqmptd 6159 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
13 | 10, 2, 4, 11, 12 | offval2 6812 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))) |
14 | 10, 4, 2, 12, 11 | offval2 6812 | . 2 ⊢ (𝜑 → (𝐺 ∘𝑓 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤)))) |
15 | 9, 13, 14 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐺 ∘𝑓 𝑅𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 |
This theorem is referenced by: plydivlem4 23855 quotcan 23868 dchrabl 24779 plymulx0 29950 lfladdcom 33377 expgrowth 37556 amgmwlem 42357 |
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