MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prodeq1d Structured version   Visualization version   GIF version

Theorem prodeq1d 14858
Description: Equality deduction for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
prodeq1d (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hints:   𝜑(𝑘)   𝐶(𝑘)

Proof of Theorem prodeq1d
StepHypRef Expression
1 prodeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 prodeq1 14846 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2syl 17 1 (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  cprod 14842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-seq 13009  df-prod 14843
This theorem is referenced by:  prodeq12dv  14863  prodeq12rdv  14864  fprodf1o  14883  prodss  14884  fprod1  14900  fprodp1  14906  fprodfac  14910  fprodabs  14911  fprod2d  14918  fprodcom2  14921  risefacval  14945  fallfacval  14946  risefacval2  14947  fallfacval2  14948  risefacp1  14966  fallfacp1  14967  fallfacval4  14980  fprodefsum  15031  prmoval  15944  prmop1  15949  prmgapprmo  15973  gausslemma2dlem4  25315  breprexplema  31048  breprexplemc  31050  breprexp  31051  circlemethhgt  31061  bcprod  31962  dvmptfprodlem  40677  dvmptfprod  40678  ovnval  41275  hoiprodp1  41322  hoidmv1le  41328  hspmbllem1  41360  fmtnorec2  41983
  Copyright terms: Public domain W3C validator