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Mirrors > Home > MPE Home > Th. List > offval | Structured version Visualization version GIF version |
Description: Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.) |
Ref | Expression |
---|---|
offval.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
offval.2 | ⊢ (𝜑 → 𝐺 Fn 𝐵) |
offval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
offval.4 | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
offval.5 | ⊢ (𝐴 ∩ 𝐵) = 𝑆 |
offval.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) |
offval.7 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) |
Ref | Expression |
---|---|
offval | ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | offval.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | fnex 7236 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | fnex 7236 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V) | |
8 | 5, 6, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
9 | 1 | fndmd 6673 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
10 | 5 | fndmd 6673 | . . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵) |
11 | 9, 10 | ineq12d 4228 | . . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
12 | offval.5 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
13 | 11, 12 | eqtrdi 2790 | . . . . 5 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆) |
14 | 13 | mpteq1d 5242 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
15 | inex1g 5324 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
16 | 12, 15 | eqeltrrid 2843 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ∈ V) |
17 | mptexg 7240 | . . . . 5 ⊢ (𝑆 ∈ V → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) | |
18 | 2, 16, 17 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
19 | 14, 18 | eqeltrd 2838 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
20 | dmeq 5916 | . . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
21 | dmeq 5916 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
22 | 20, 21 | ineqan12d 4229 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺)) |
23 | fveq1 6905 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
24 | fveq1 6905 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
25 | 23, 24 | oveqan12d 7449 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
26 | 22, 25 | mpteq12dv 5238 | . . . 4 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
27 | df-of 7696 | . . . 4 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
28 | 26, 27 | ovmpoga 7586 | . . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
29 | 4, 8, 19, 28 | syl3anc 1370 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
30 | 12 | eleq2i 2830 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝑆) |
31 | elin 3978 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
32 | 30, 31 | bitr3i 277 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
33 | offval.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | |
34 | 33 | adantrr 717 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = 𝐶) |
35 | offval.7 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) | |
36 | 35 | adantrl 716 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐺‘𝑥) = 𝐷) |
37 | 34, 36 | oveq12d 7448 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷)) |
38 | 32, 37 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷)) |
39 | 38 | mpteq2dva 5247 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
40 | 29, 14, 39 | 3eqtrd 2778 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 ↦ cmpt 5230 dom cdm 5688 Fn wfn 6557 ‘cfv 6562 (class class class)co 7430 ∘f cof 7694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 |
This theorem is referenced by: ofval 7707 offn 7709 offval2f 7711 off 7714 ofres 7715 offval2 7716 coof 7720 ofco 7721 offveqb 7723 suppssof1 8222 o1rlimmul 15651 frlmipval 21816 frlmphllem 21817 frlmphl 21818 gsumbagdiaglem 21967 psrascl 22016 evlslem1 22123 mhpmulcl 22170 psdmplcl 22183 psdadd 22184 psdmul 22187 psrplusgpropd 22252 evls1fpws 22388 mat1dimscm 22496 rrxcph 25439 rrxds 25440 mbfadd 25709 mbfsub 25710 mbfmullem2 25773 mbfmul 25775 bddmulibl 25888 dvcmulf 25996 ofrn2 32656 off2 32657 ofresid 32658 islinds5 33374 ellspds 33375 ply1gsumz 33598 ofcof 34087 plymul02 34539 signsplypnf 34543 signsply0 34544 matunitlindflem1 37602 matunitlindflem2 37603 poimirlem3 37609 poimirlem4 37610 poimirlem16 37622 poimirlem19 37625 poimirlem28 37634 broucube 37640 itg2addnc 37660 ftc1anclem8 37686 evlsvvval 42549 dflinc2 48255 fdivmpt 48389 |
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