Theorem prodeq1 14975

Description: Equality theorem for a product. (Contributed by Scott Fenton, 1-Dec-2017.)

Assertion

Ref	Expression
prodeq1	⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Distinct variable groups:   𝐴,𝑘   𝐵,𝑘

Allowed substitution hint:   𝐶(𝑘)

Proof of Theorem prodeq1

Step	Hyp	Ref	Expression
1		nfcv 2942	. 2 ⊢ Ⅎ𝑘𝐴
2		nfcv 2942	. 2 ⊢ Ⅎ𝑘𝐵
3	1, 2	prodeq1f 14974	1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1653  ∏cprod 14971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-xp 5319  df-cnv 5321  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-iota 6065  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-seq 13055  df-prod 14972

This theorem is referenced by:  prodeq1i  14984  prodeq1d  14987  prod1  15010  fprodf1o  15012  fprodss  15014  fprodcllem  15017  fprodmul  15026  fproddiv  15027  fprodconst  15044  fprodn0  15045  fprod2d  15047  fprodmodd  15063  coprmprod  15708  coprmproddvds  15710  fprodexp  40565  fprodabs2  40566  mccl  40569  fprodcn  40571  fprodcncf  40853  dvmptfprod  40899  dvnprodlem3  40902  hoidmvval  41532  ovnhoi  41558  hspmbllem2  41582  fmtnorec2  42232