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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 7172 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 585 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | fnex 7172 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V) | |
8 | 5, 6, 7 | syl2anc 585 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
9 | 1 | fndmd 6612 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
10 | 5 | fndmd 6612 | . . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵) |
11 | 9, 10 | ineq12d 4178 | . . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
12 | offval.5 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
13 | 11, 12 | eqtrdi 2793 | . . . . 5 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆) |
14 | 13 | mpteq1d 5205 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
15 | inex1g 5281 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
16 | 12, 15 | eqeltrrid 2843 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ∈ V) |
17 | mptexg 7176 | . . . . 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 5864 | . . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
21 | dmeq 5864 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
22 | 20, 21 | ineqan12d 4179 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺)) |
23 | fveq1 6846 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
24 | fveq1 6846 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
25 | 23, 24 | oveqan12d 7381 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
26 | 22, 25 | mpteq12dv 5201 | . . . 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 7622 | . . . 4 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
28 | 26, 27 | ovmpoga 7514 | . . 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 1372 | . 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 3931 | . . . . 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 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 7380 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷)) |
38 | 32, 37 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝐶𝑅𝐷)) |
39 | 38 | mpteq2dva 5210 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
40 | 29, 14, 39 | 3eqtrd 2781 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∩ cin 3914 ↦ cmpt 5193 dom cdm 5638 Fn wfn 6496 ‘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: ofval 7633 offn 7635 offval2f 7637 off 7640 ofres 7641 offval2 7642 ofco 7645 offveqb 7647 suppssof1 8135 o1rlimmul 15508 frlmipval 21201 frlmphllem 21202 frlmphl 21203 gsumbagdiaglemOLD 21356 gsumbagdiaglem 21359 evlslem1 21508 mhpmulcl 21555 psrplusgpropd 21623 mat1dimscm 21840 rrxcph 24772 rrxds 24773 mbfadd 25041 mbfsub 25042 mbfmullem2 25105 mbfmul 25107 bddmulibl 25219 dvcmulf 25325 ofrn2 31598 off2 31599 ofresid 31600 islinds5 32196 ellspds 32197 evls1fpws 32311 ofcof 32746 plymul02 33198 signsplypnf 33202 signsply0 33203 matunitlindflem1 36103 matunitlindflem2 36104 poimirlem3 36110 poimirlem4 36111 poimirlem16 36123 poimirlem19 36126 poimirlem28 36135 broucube 36141 itg2addnc 36161 ftc1anclem8 36187 evlsbagval 40777 mhphf 40800 dflinc2 46565 fdivmpt 46700 |
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