MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prodeq1 Structured version   Visualization version   GIF version

Theorem prodeq1 15251
Description: Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)
Assertion
Ref Expression
prodeq1 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1
StepHypRef Expression
1 nfcv 2974 . 2 𝑘𝐴
2 nfcv 2974 . 2 𝑘𝐵
31, 2prodeq1f 15250 1 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cprod 15247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seq 13358  df-prod 15248
This theorem is referenced by:  prodeq1i  15260  prodeq1d  15263  prod1  15286  fprodf1o  15288  fprodss  15290  fprodcllem  15293  fprodmul  15302  fproddiv  15303  fprodconst  15320  fprodn0  15321  fprod2d  15323  fprodmodd  15339  coprmprod  15993  coprmproddvds  15995  fprodexp  41751  fprodabs2  41752  mccl  41755  fprodcn  41757  fprodcncf  42060  dvmptfprod  42106  dvnprodlem3  42109  hoidmvval  42736  ovnhoi  42762  hspmbllem2  42786  fmtnorec2  43582
  Copyright terms: Public domain W3C validator