prodeq1i - Metamath Proof Explorer

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Theorem prodeq1i 15895

Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017.)

**Hypothesis**
Ref	Expression
prodeq1i.1	⊢ 𝐴 = 𝐵

**Assertion**
Ref	Expression
prodeq1i	⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 Allowed substitution hint: 𝐶(𝑘)

**Proof of Theorem prodeq1i**
Step	Hyp	Ref	Expression
1		prodeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		prodeq1 15886	. 2 ⊢ (𝐴 = 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)
3	1, 2	ax-mp 5	1 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∏cprod 15882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5684 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-iota 6500 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-seq 14000 df-prod 15883

This theorem is referenced by: prodeq12i 15897 fprodxp 15959 risefac0 16004 fallfacfwd 16013 prmo0 17005 breprexp 34265 etransclem31 45653 etransclem35 45657 hoidmv1le 45982 fmtnorec2 46883

	
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