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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 2611 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2611 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 6802 | . . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | 8 | fveq1d 6105 | . . 3 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
10 | 9 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
11 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
12 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
13 | 11, 12 | oveq12d 6567 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
14 | eqid 2610 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
15 | ovex 6577 | . . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6191 | . . 3 ⊢ (𝑋 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
17 | 16 | adantl 481 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
18 | inss1 3795 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
19 | 5, 18 | eqsstr3i 3599 | . . . . 5 ⊢ 𝑆 ⊆ 𝐴 |
20 | 19 | sseli 3564 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴) |
21 | ofval.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
22 | 20, 21 | sylan2 490 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
23 | inss2 3796 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
24 | 5, 23 | eqsstr3i 3599 | . . . . 5 ⊢ 𝑆 ⊆ 𝐵 |
25 | 24 | sseli 3564 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵) |
26 | ofval.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) | |
27 | 25, 26 | sylan2 490 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
28 | 22, 27 | oveq12d 6567 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷)) |
29 | 10, 17, 28 | 3eqtrd 2648 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ↦ cmpt 4643 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 |
This theorem is referenced by: fnfvof 6809 offveq 6816 ofc1 6818 ofc2 6819 suppofss1d 7219 suppofss2d 7220 ofsubeq0 10894 ofnegsub 10895 ofsubge0 10896 seqof 12720 o1of2 14191 gsumzaddlem 18144 psrbagcon 19192 psrbagconf1o 19195 psrdi 19227 psrdir 19228 mplsubglem 19255 matplusgcell 20058 matsubgcell 20059 rrxcph 22988 mbfaddlem 23233 i1faddlem 23266 i1fmullem 23267 itg1lea 23285 mbfi1flimlem 23295 itg2split 23322 itg2monolem1 23323 itg2addlem 23331 dvaddbr 23507 dvmulbr 23508 plyaddlem1 23773 coeeulem 23784 coeaddlem 23809 dgradd2 23828 dgrcolem2 23834 ofmulrt 23841 plydivlem3 23854 plydivlem4 23855 plydiveu 23857 plyrem 23864 vieta1lem2 23870 elqaalem3 23880 qaa 23882 basellem7 24613 basellem9 24615 poimirlem1 32580 poimirlem2 32581 poimirlem6 32585 poimirlem7 32586 poimirlem10 32589 poimirlem11 32590 poimirlem12 32591 poimirlem17 32596 poimirlem20 32599 poimirlem23 32602 poimirlem29 32608 poimirlem31 32610 poimirlem32 32611 broucube 32613 itg2addnclem3 32633 itg2addnc 32634 ftc1anclem5 32659 lfladdcl 33376 ldualvaddval 33436 dgrsub2 36724 mpaaeu 36739 caofcan 37544 ofmul12 37546 ofdivrec 37547 ofdivcan4 37548 ofdivdiv2 37549 binomcxplemrat 37571 binomcxplemnotnn0 37577 mndpsuppss 41946 amgmwlem 42357 |
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