Theorem prodeq1 15263

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 2977	. 2 ⊢ Ⅎ𝑘𝐴
2		nfcv 2977	. 2 ⊢ Ⅎ𝑘𝐵
3	1, 2	prodeq1f 15262	1 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   = wceq 1537  ∏cprod 15259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-iota 6314  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-seq 13371  df-prod 15260

This theorem is referenced by:  prodeq1i  15272  prodeq1d  15275  prod1  15298  fprodf1o  15300  fprodss  15302  fprodcllem  15305  fprodmul  15314  fproddiv  15315  fprodconst  15332  fprodn0  15333  fprod2d  15335  fprodmodd  15351  coprmprod  16005  coprmproddvds  16007  fprodexp  41895  fprodabs2  41896  mccl  41899  fprodcn  41901  fprodcncf  42204  dvmptfprod  42250  dvnprodlem3  42253  hoidmvval  42879  ovnhoi  42905  hspmbllem2  42929  fmtnorec2  43725