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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 6957 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | fnex 6957 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V) | |
8 | 5, 6, 7 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
9 | 1 | fndmd 6427 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
10 | 5 | fndmd 6427 | . . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵) |
11 | 9, 10 | ineq12d 4140 | . . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
12 | offval.5 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
13 | 11, 12 | eqtrdi 2849 | . . . . 5 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆) |
14 | 13 | mpteq1d 5119 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
15 | inex1g 5187 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
16 | 12, 15 | eqeltrrid 2895 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ∈ V) |
17 | mptexg 6961 | . . . . 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 2890 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
20 | dmeq 5736 | . . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
21 | dmeq 5736 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
22 | 20, 21 | ineqan12d 4141 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺)) |
23 | fveq1 6644 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
24 | fveq1 6644 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
25 | 23, 24 | oveqan12d 7154 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
26 | 22, 25 | mpteq12dv 5115 | . . . 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 7389 | . . . 4 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
28 | 26, 27 | ovmpoga 7283 | . . 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 1368 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
30 | 12 | eleq2i 2881 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝑆) |
31 | elin 3897 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
32 | 30, 31 | bitr3i 280 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
33 | offval.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | |
34 | 33 | adantrr 716 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = 𝐶) |
35 | offval.7 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) | |
36 | 35 | adantrl 715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐺‘𝑥) = 𝐷) |
37 | 34, 36 | oveq12d 7153 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷)) |
38 | 32, 37 | sylan2b 596 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷)) |
39 | 38 | mpteq2dva 5125 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
40 | 29, 14, 39 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ↦ cmpt 5110 dom cdm 5519 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∘f cof 7387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 |
This theorem is referenced by: ofval 7398 offn 7400 offval2f 7401 off 7404 ofres 7405 offval2 7406 ofco 7409 offveqb 7411 suppssof1 7846 o1rlimmul 14967 frlmipval 20468 frlmphllem 20469 frlmphl 20470 gsumbagdiaglem 20613 evlslem1 20754 psrplusgpropd 20865 mat1dimscm 21080 rrxcph 23996 rrxds 23997 mbfadd 24265 mbfsub 24266 mbfmullem2 24328 mbfmul 24330 bddmulibl 24442 dvcmulf 24548 ofrn2 30401 off2 30402 ofresid 30403 islinds5 30983 ellspds 30984 ofcof 31476 plymul02 31926 signsplypnf 31930 signsply0 31931 matunitlindflem1 35053 matunitlindflem2 35054 poimirlem3 35060 poimirlem4 35061 poimirlem16 35073 poimirlem19 35076 poimirlem28 35085 broucube 35091 itg2addnc 35111 ftc1anclem8 35137 dflinc2 44819 fdivmpt 44954 |
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