MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofval Structured version   Visualization version   GIF version

Theorem ofval 7412
Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)
Hypotheses
Ref Expression
offval.1 (𝜑𝐹 Fn 𝐴)
offval.2 (𝜑𝐺 Fn 𝐵)
offval.3 (𝜑𝐴𝑉)
offval.4 (𝜑𝐵𝑊)
offval.5 (𝐴𝐵) = 𝑆
ofval.6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
ofval.7 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
Assertion
Ref Expression
ofval ((𝜑𝑋𝑆) → ((𝐹f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))

Proof of Theorem ofval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . 5 (𝜑𝐹 Fn 𝐴)
2 offval.2 . . . . 5 (𝜑𝐺 Fn 𝐵)
3 offval.3 . . . . 5 (𝜑𝐴𝑉)
4 offval.4 . . . . 5 (𝜑𝐵𝑊)
5 offval.5 . . . . 5 (𝐴𝐵) = 𝑆
6 eqidd 2822 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2822 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 7410 . . . 4 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
98fveq1d 6667 . . 3 (𝜑 → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
109adantr 483 . 2 ((𝜑𝑋𝑆) → ((𝐹f 𝑅𝐺)‘𝑋) = ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋))
11 fveq2 6665 . . . . 5 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
12 fveq2 6665 . . . . 5 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1311, 12oveq12d 7168 . . . 4 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑋)𝑅(𝐺𝑋)))
14 eqid 2821 . . . 4 (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))
15 ovex 7183 . . . 4 ((𝐹𝑋)𝑅(𝐺𝑋)) ∈ V
1613, 14, 15fvmpt 6763 . . 3 (𝑋𝑆 → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
1716adantl 484 . 2 ((𝜑𝑋𝑆) → ((𝑥𝑆 ↦ ((𝐹𝑥)𝑅(𝐺𝑥)))‘𝑋) = ((𝐹𝑋)𝑅(𝐺𝑋)))
18 inss1 4205 . . . . . 6 (𝐴𝐵) ⊆ 𝐴
195, 18eqsstrri 4002 . . . . 5 𝑆𝐴
2019sseli 3963 . . . 4 (𝑋𝑆𝑋𝐴)
21 ofval.6 . . . 4 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2220, 21sylan2 594 . . 3 ((𝜑𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 4206 . . . . . 6 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstrri 4002 . . . . 5 𝑆𝐵
2524sseli 3963 . . . 4 (𝑋𝑆𝑋𝐵)
26 ofval.7 . . . 4 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2725, 26sylan2 594 . . 3 ((𝜑𝑋𝑆) → (𝐺𝑋) = 𝐷)
2822, 27oveq12d 7168 . 2 ((𝜑𝑋𝑆) → ((𝐹𝑋)𝑅(𝐺𝑋)) = (𝐶𝑅𝐷))
2910, 17, 283eqtrd 2860 1 ((𝜑𝑋𝑆) → ((𝐹f 𝑅𝐺)‘𝑋) = (𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  cin 3935  cmpt 5139   Fn wfn 6345  cfv 6350  (class class class)co 7150  f cof 7401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403
This theorem is referenced by:  fnfvof  7417  offveq  7424  ofc1  7426  ofc2  7427  suppofss1d  7862  suppofss2d  7863  ofsubeq0  11629  ofnegsub  11630  ofsubge0  11631  seqof  13421  o1of2  14963  gsumzaddlem  19035  psrbagcon  20145  psrbagconf1o  20148  psrdi  20180  psrdir  20181  mplsubglem  20208  matplusgcell  21036  matsubgcell  21037  rrxcph  23989  mbfaddlem  24255  i1faddlem  24288  i1fmullem  24289  itg1lea  24307  mbfi1flimlem  24317  itg2split  24344  itg2monolem1  24345  itg2addlem  24353  dvaddbr  24529  dvmulbr  24530  plyaddlem1  24797  coeeulem  24808  coeaddlem  24833  dgradd2  24852  dgrcolem2  24858  ofmulrt  24865  plydivlem3  24878  plydivlem4  24879  plydiveu  24881  plyrem  24888  vieta1lem2  24894  elqaalem3  24904  qaa  24906  basellem7  25658  basellem9  25660  circlemethhgt  31909  poimirlem1  34887  poimirlem2  34888  poimirlem6  34892  poimirlem7  34893  poimirlem10  34896  poimirlem11  34897  poimirlem12  34898  poimirlem17  34903  poimirlem20  34906  poimirlem23  34909  poimirlem29  34915  poimirlem31  34917  poimirlem32  34918  broucube  34920  itg2addnclem3  34939  itg2addnc  34940  ftc1anclem5  34965  lfladdcl  36201  ldualvaddval  36261  dgrsub2  39728  mpaaeu  39743  caofcan  40648  ofmul12  40650  ofdivrec  40651  ofdivcan4  40652  ofdivdiv2  40653  binomcxplemrat  40675  binomcxplemnotnn0  40681  mndpsuppss  44412  amgmwlem  44896
  Copyright terms: Public domain W3C validator