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Theorem prodeq1i 14487
 Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
prodeq1i.1 𝐴 = 𝐵
Assertion
Ref Expression
prodeq1i 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘
Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1i
StepHypRef Expression
1 prodeq1i.1 . 2 𝐴 = 𝐵
2 prodeq1 14478 . 2 (𝐴 = 𝐵 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶)
31, 2ax-mp 5 1 𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐶
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-prod 14475 This theorem is referenced by:  prodeq12i  14489  fprodxp  14551  risefac0  14597  fallfacfwd  14606  prmo0  15578  etransclem31  39158  etransclem35  39162  hoidmv1le  39484  fmtnorec2  39993
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