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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 6983 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
4 | 1, 2, 3 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
5 | offval.2 | . . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
6 | offval.4 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
7 | fnex 6983 | . . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V) | |
8 | 5, 6, 7 | syl2anc 586 | . . 3 ⊢ (𝜑 → 𝐺 ∈ V) |
9 | 1 | fndmd 6459 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
10 | 5 | fndmd 6459 | . . . . . . 7 ⊢ (𝜑 → dom 𝐺 = 𝐵) |
11 | 9, 10 | ineq12d 4193 | . . . . . 6 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵)) |
12 | offval.5 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
13 | 11, 12 | syl6eq 2875 | . . . . 5 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆) |
14 | 13 | mpteq1d 5158 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
15 | inex1g 5226 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∩ 𝐵) ∈ V) | |
16 | 12, 15 | eqeltrrid 2921 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝑆 ∈ V) |
17 | mptexg 6987 | . . . . 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 2916 | . . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) |
20 | dmeq 5775 | . . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹) | |
21 | dmeq 5775 | . . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺) | |
22 | 20, 21 | ineqan12d 4194 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺)) |
23 | fveq1 6672 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
24 | fveq1 6672 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
25 | 23, 24 | oveqan12d 7178 | . . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) |
26 | 22, 25 | mpteq12dv 5154 | . . . 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 7412 | . . . 4 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) | |
28 | 26, 27 | ovmpoga 7307 | . . 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 1367 | . 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
30 | 12 | eleq2i 2907 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ 𝑆) |
31 | elin 4172 | . . . . 5 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
32 | 30, 31 | bitr3i 279 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) |
33 | offval.6 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶) | |
34 | 33 | adantrr 715 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = 𝐶) |
35 | offval.7 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷) | |
36 | 35 | adantrl 714 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) → (𝐺‘𝑥) = 𝐷) |
37 | 34, 36 | oveq12d 7177 | . . . 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 5164 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
40 | 29, 14, 39 | 3eqtrd 2863 | 1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ (𝐶𝑅𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∩ cin 3938 ↦ cmpt 5149 dom cdm 5558 Fn wfn 6353 ‘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-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: ofval 7421 offn 7423 offval2f 7424 off 7427 ofres 7428 offval2 7429 ofco 7432 offveqb 7434 suppssof1 7866 o1rlimmul 14978 gsumbagdiaglem 20158 evlslem1 20298 psrplusgpropd 20407 frlmipval 20926 frlmphllem 20927 frlmphl 20928 mat1dimscm 21087 rrxcph 23998 rrxds 23999 mbfadd 24265 mbfsub 24266 mbfmullem2 24328 mbfmul 24330 bddmulibl 24442 dvcmulf 24545 ofrn2 30390 off2 30391 ofresid 30392 islinds5 30936 ellspds 30937 ofcof 31370 plymul02 31820 signsplypnf 31824 signsply0 31825 matunitlindflem1 34892 matunitlindflem2 34893 poimirlem3 34899 poimirlem4 34900 poimirlem16 34912 poimirlem19 34915 poimirlem28 34924 broucube 34930 itg2addnc 34950 ftc1anclem8 34978 dflinc2 44472 fdivmpt 44607 |
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