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

Theorem prodeq1d 14857
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 14845 . 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 1630  cprod 14841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  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 6795  df-oprab 6796  df-mpt2 6797  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seq 13008  df-prod 14842
This theorem is referenced by:  prodeq12dv  14862  prodeq12rdv  14863  fprodf1o  14882  prodss  14883  fprod1  14899  fprodp1  14905  fprodfac  14909  fprodabs  14910  fprod2d  14917  fprodcom2  14920  risefacval  14944  fallfacval  14945  risefacval2  14946  fallfacval2  14947  risefacp1  14965  fallfacp1  14966  fallfacval4  14979  fprodefsum  15030  prmoval  15943  prmop1  15948  prmgapprmo  15972  gausslemma2dlem4  25314  breprexplema  31042  breprexplemc  31044  breprexp  31045  circlemethhgt  31055  bcprod  31956  dvmptfprodlem  40671  dvmptfprod  40672  ovnval  41269  hoiprodp1  41316  hoidmv1le  41322  hspmbllem1  41354  fmtnorec2  41973
  Copyright terms: Public domain W3C validator