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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 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝑆) |
3 | caofcom.3 | . . . . . 6 ⊢ (𝜑 → 𝐺:𝐴⟶𝑆) | |
4 | 3 | ffvelrnda 6854 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐺‘𝑤) ∈ 𝑆) |
5 | 2, 4 | jca 514 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) |
6 | caofcom.4 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝑅𝑦) = (𝑦𝑅𝑥)) | |
7 | 6 | caovcomg 7346 | . . . 4 ⊢ ((𝜑 ∧ ((𝐹‘𝑤) ∈ 𝑆 ∧ (𝐺‘𝑤) ∈ 𝑆)) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) |
8 | 5, 7 | syldan 593 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑤)𝑅(𝐺‘𝑤)) = ((𝐺‘𝑤)𝑅(𝐹‘𝑤))) |
9 | 8 | mpteq2dva 5164 | . 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 6736 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑤 ∈ 𝐴 ↦ (𝐹‘𝑤))) |
12 | 3 | feqmptd 6736 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑤 ∈ 𝐴 ↦ (𝐺‘𝑤))) |
13 | 10, 2, 4, 11, 12 | offval2 7429 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑤 ∈ 𝐴 ↦ ((𝐹‘𝑤)𝑅(𝐺‘𝑤)))) |
14 | 10, 4, 2, 12, 11 | offval2 7429 | . 2 ⊢ (𝜑 → (𝐺 ∘f 𝑅𝐹) = (𝑤 ∈ 𝐴 ↦ ((𝐺‘𝑤)𝑅(𝐹‘𝑤)))) |
15 | 9, 13, 14 | 3eqtr4d 2869 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐺 ∘f 𝑅𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ↦ cmpt 5149 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∘f cof 7410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 |
This theorem is referenced by: plydivlem4 24888 quotcan 24901 dchrabl 25833 plymulx0 31821 lfladdcom 36212 expgrowth 40673 amgmwlem 44910 |
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