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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 2799 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2799 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 7396 | . . . 4 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | 8 | fveq1d 6647 | . . 3 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
10 | 9 | adantr 484 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
11 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
12 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
13 | 11, 12 | oveq12d 7153 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
14 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
15 | ovex 7168 | . . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6745 | . . 3 ⊢ (𝑋 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
17 | 16 | adantl 485 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
18 | inss1 4155 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
19 | 5, 18 | eqsstrri 3950 | . . . . 5 ⊢ 𝑆 ⊆ 𝐴 |
20 | 19 | sseli 3911 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴) |
21 | ofval.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
22 | 20, 21 | sylan2 595 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
23 | inss2 4156 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
24 | 5, 23 | eqsstrri 3950 | . . . . 5 ⊢ 𝑆 ⊆ 𝐵 |
25 | 24 | sseli 3911 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵) |
26 | ofval.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) | |
27 | 25, 26 | sylan2 595 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
28 | 22, 27 | oveq12d 7153 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷)) |
29 | 10, 17, 28 | 3eqtrd 2837 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ↦ cmpt 5110 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: fnfvof 7403 offveq 7410 ofc1 7412 ofc2 7413 suppofss1d 7851 suppofss2d 7852 ofsubeq0 11622 ofnegsub 11623 ofsubge0 11624 seqof 13423 o1of2 14961 gsumzaddlem 19034 psrbagcon 20609 psrbagconf1o 20612 psrdi 20644 psrdir 20645 mplsubglem 20672 matplusgcell 21038 matsubgcell 21039 rrxcph 23996 mbfaddlem 24264 i1faddlem 24297 i1fmullem 24298 itg1lea 24316 mbfi1flimlem 24326 itg2split 24353 itg2monolem1 24354 itg2addlem 24362 dvaddbr 24541 dvmulbr 24542 plyaddlem1 24810 coeeulem 24821 coeaddlem 24846 dgradd2 24865 dgrcolem2 24871 ofmulrt 24878 plydivlem3 24891 plydivlem4 24892 plydiveu 24894 plyrem 24901 vieta1lem2 24907 elqaalem3 24917 qaa 24919 basellem7 25672 basellem9 25674 circlemethhgt 32024 poimirlem1 35058 poimirlem2 35059 poimirlem6 35063 poimirlem7 35064 poimirlem10 35067 poimirlem11 35068 poimirlem12 35069 poimirlem17 35074 poimirlem20 35077 poimirlem23 35080 poimirlem29 35086 poimirlem31 35088 poimirlem32 35089 broucube 35091 itg2addnclem3 35110 itg2addnc 35111 ftc1anclem5 35134 lfladdcl 36367 ldualvaddval 36427 ofun 39416 fsuppind 39456 dgrsub2 40079 mpaaeu 40094 caofcan 41027 ofmul12 41029 ofdivrec 41030 ofdivcan4 41031 ofdivdiv2 41032 binomcxplemrat 41054 binomcxplemnotnn0 41060 mndpsuppss 44773 amgmwlem 45330 |
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