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Theorem prodeq1 15258
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 2982 . 2 𝑘𝐴
2 nfcv 2982 . 2 𝑘𝐵
31, 2prodeq1f 15257 1 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  cprod 15254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-iota 6313  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seq 13365  df-prod 15255
This theorem is referenced by:  prodeq1i  15267  prodeq1d  15270  prod1  15293  fprodf1o  15295  fprodss  15297  fprodcllem  15300  fprodmul  15309  fproddiv  15310  fprodconst  15327  fprodn0  15328  fprod2d  15330  fprodmodd  15346  coprmprod  16000  coprmproddvds  16002  fprodexp  41759  fprodabs2  41760  mccl  41763  fprodcn  41765  fprodcncf  42068  dvmptfprod  42114  dvnprodlem3  42117  hoidmvval  42744  ovnhoi  42770  hspmbllem2  42794  fmtnorec2  43556
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