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Mirrors > Home > MPE Home > Th. List > ofval | Structured version Visualization version GIF version |
Description: Evaluate a function operation at a point. (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 |
ofval.6 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) |
ofval.7 | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) |
Ref | Expression |
---|---|
ofval | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | offval.1 | . . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
2 | offval.2 | . . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐵) | |
3 | offval.3 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | offval.4 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
5 | offval.5 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) = 𝑆 | |
6 | eqidd 2824 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2824 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 7418 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | 8 | fveq1d 6674 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
10 | 9 | adantr 483 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
11 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
12 | fveq2 6672 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
13 | 11, 12 | oveq12d 7176 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
14 | eqid 2823 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
15 | ovex 7191 | . . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6770 | . . 3 ⊢ (𝑋 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
17 | 16 | adantl 484 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
18 | inss1 4207 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
19 | 5, 18 | eqsstrri 4004 | . . . . 5 ⊢ 𝑆 ⊆ 𝐴 |
20 | 19 | sseli 3965 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴) |
21 | ofval.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
22 | 20, 21 | sylan2 594 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
23 | inss2 4208 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
24 | 5, 23 | eqsstrri 4004 | . . . . 5 ⊢ 𝑆 ⊆ 𝐵 |
25 | 24 | sseli 3965 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵) |
26 | ofval.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) | |
27 | 25, 26 | sylan2 594 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
28 | 22, 27 | oveq12d 7176 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷)) |
29 | 10, 17, 28 | 3eqtrd 2862 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ↦ cmpt 5148 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 ∘f cof 7409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 |
This theorem is referenced by: fnfvof 7425 offveq 7432 ofc1 7434 ofc2 7435 suppofss1d 7870 suppofss2d 7871 ofsubeq0 11637 ofnegsub 11638 ofsubge0 11639 seqof 13430 o1of2 14971 gsumzaddlem 19043 psrbagcon 20153 psrbagconf1o 20156 psrdi 20188 psrdir 20189 mplsubglem 20216 matplusgcell 21044 matsubgcell 21045 rrxcph 23997 mbfaddlem 24263 i1faddlem 24296 i1fmullem 24297 itg1lea 24315 mbfi1flimlem 24325 itg2split 24352 itg2monolem1 24353 itg2addlem 24361 dvaddbr 24537 dvmulbr 24538 plyaddlem1 24805 coeeulem 24816 coeaddlem 24841 dgradd2 24860 dgrcolem2 24866 ofmulrt 24873 plydivlem3 24886 plydivlem4 24887 plydiveu 24889 plyrem 24896 vieta1lem2 24902 elqaalem3 24912 qaa 24914 basellem7 25666 basellem9 25668 circlemethhgt 31916 poimirlem1 34895 poimirlem2 34896 poimirlem6 34900 poimirlem7 34901 poimirlem10 34904 poimirlem11 34905 poimirlem12 34906 poimirlem17 34911 poimirlem20 34914 poimirlem23 34917 poimirlem29 34923 poimirlem31 34925 poimirlem32 34926 broucube 34928 itg2addnclem3 34947 itg2addnc 34948 ftc1anclem5 34973 lfladdcl 36209 ldualvaddval 36269 dgrsub2 39742 mpaaeu 39757 caofcan 40662 ofmul12 40664 ofdivrec 40665 ofdivcan4 40666 ofdivdiv2 40667 binomcxplemrat 40689 binomcxplemnotnn0 40695 mndpsuppss 44426 amgmwlem 44910 |
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