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

Theorem prodeq1 14633
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 2763 . 2 𝑘𝐴
2 nfcv 2763 . 2 𝑘𝐵
31, 2prodeq1f 14632 1 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  cprod 14629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-iota 5849  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-seq 12797  df-prod 14630
This theorem is referenced by:  prodeq1i  14642  prodeq1d  14645  prod1  14668  fprodf1o  14670  fprodss  14672  fprodcllem  14675  fprodmul  14684  fproddiv  14685  fprodconst  14702  fprodn0  14703  fprod2d  14705  fprodmodd  14722  coprmprod  15369  coprmproddvds  15371  fprodexp  39632  fprodabs2  39633  mccl  39636  fprodcn  39638  fprodcncf  39883  dvmptfprod  39929  dvnprodlem3  39932  hoidmvval  40560  ovnhoi  40586  hspmbllem2  40610  fmtnorec2  41226
  Copyright terms: Public domain W3C validator