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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 2761 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥)) | |
7 | eqidd 2761 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥)) | |
8 | 1, 2, 3, 4, 5, 6, 7 | offval 7069 | . . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) |
9 | 8 | fveq1d 6354 | . . 3 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
10 | 9 | adantr 472 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋)) |
11 | fveq2 6352 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
12 | fveq2 6352 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
13 | 11, 12 | oveq12d 6831 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
14 | eqid 2760 | . . . 4 ⊢ (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) | |
15 | ovex 6841 | . . . 4 ⊢ ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) ∈ V | |
16 | 13, 14, 15 | fvmpt 6444 | . . 3 ⊢ (𝑋 ∈ 𝑆 → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
17 | 16 | adantl 473 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋))) |
18 | inss1 3976 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
19 | 5, 18 | eqsstr3i 3777 | . . . . 5 ⊢ 𝑆 ⊆ 𝐴 |
20 | 19 | sseli 3740 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐴) |
21 | ofval.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶) | |
22 | 20, 21 | sylan2 492 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶) |
23 | inss2 3977 | . . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
24 | 5, 23 | eqsstr3i 3777 | . . . . 5 ⊢ 𝑆 ⊆ 𝐵 |
25 | 24 | sseli 3740 | . . . 4 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝐵) |
26 | ofval.7 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷) | |
27 | 25, 26 | sylan2 492 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷) |
28 | 22, 27 | oveq12d 6831 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹‘𝑋)𝑅(𝐺‘𝑋)) = (𝐶𝑅𝐷)) |
29 | 10, 17, 28 | 3eqtrd 2798 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑆) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∩ cin 3714 ↦ cmpt 4881 Fn wfn 6044 ‘cfv 6049 (class class class)co 6813 ∘𝑓 cof 7060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 |
This theorem is referenced by: fnfvof 7076 offveq 7083 ofc1 7085 ofc2 7086 suppofss1d 7501 suppofss2d 7502 ofsubeq0 11209 ofnegsub 11210 ofsubge0 11211 seqof 13052 o1of2 14542 gsumzaddlem 18521 psrbagcon 19573 psrbagconf1o 19576 psrdi 19608 psrdir 19609 mplsubglem 19636 matplusgcell 20441 matsubgcell 20442 rrxcph 23380 mbfaddlem 23626 i1faddlem 23659 i1fmullem 23660 itg1lea 23678 mbfi1flimlem 23688 itg2split 23715 itg2monolem1 23716 itg2addlem 23724 dvaddbr 23900 dvmulbr 23901 plyaddlem1 24168 coeeulem 24179 coeaddlem 24204 dgradd2 24223 dgrcolem2 24229 ofmulrt 24236 plydivlem3 24249 plydivlem4 24250 plydiveu 24252 plyrem 24259 vieta1lem2 24265 elqaalem3 24275 qaa 24277 basellem7 25012 basellem9 25014 circlemethhgt 31030 poimirlem1 33723 poimirlem2 33724 poimirlem6 33728 poimirlem7 33729 poimirlem10 33732 poimirlem11 33733 poimirlem12 33734 poimirlem17 33739 poimirlem20 33742 poimirlem23 33745 poimirlem29 33751 poimirlem31 33753 poimirlem32 33754 broucube 33756 itg2addnclem3 33776 itg2addnc 33777 ftc1anclem5 33802 lfladdcl 34861 ldualvaddval 34921 dgrsub2 38207 mpaaeu 38222 caofcan 39024 ofmul12 39026 ofdivrec 39027 ofdivcan4 39028 ofdivdiv2 39029 binomcxplemrat 39051 binomcxplemnotnn0 39057 mndpsuppss 42662 amgmwlem 43061 |
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