Theorem prodeq1 15324

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 2919	. 2 ⊢ Ⅎ𝑘𝐴
2		nfcv 2919	. 2 ⊢ Ⅎ𝑘𝐵
3	1, 2	prodeq1f 15323	1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1538  ∏cprod 15320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-iota 6299  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-seq 13432  df-prod 15321

This theorem is referenced by:  prodeq1i  15333  prodeq1d  15336  prod1  15359  fprodf1o  15361  fprodss  15363  fprodcllem  15366  fprodmul  15375  fproddiv  15376  fprodconst  15393  fprodn0  15394  fprod2d  15396  fprodmodd  15412  coprmprod  16070  coprmproddvds  16072  fprodexp  42637  fprodabs2  42638  mccl  42641  fprodcn  42643  fprodcncf  42943  dvmptfprod  42988  dvnprodlem3  42991  hoidmvval  43617  ovnhoi  43643  hspmbllem2  43667  fmtnorec2  44477