dfpprod2 - Mathbox for Scott Fenton

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Theorem dfpprod2 31159

Description: Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.)

**Assertion**
Ref	Expression
dfpprod2	⊢ pprod(𝐴, 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ (𝐴 ∘ (1^st ↾ (V × V)))) ∩ (^◡(2^nd ↾ (V × V)) ∘ (𝐵 ∘ (2^nd ↾ (V × V)))))

**Proof of Theorem dfpprod2**
Step	Hyp	Ref	Expression
1		df-pprod 31131	. 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1^st ↾ (V × V))) ⊗ (𝐵 ∘ (2^nd ↾ (V × V))))
2		df-txp 31130	. 2 ⊢ ((𝐴 ∘ (1^st ↾ (V × V))) ⊗ (𝐵 ∘ (2^nd ↾ (V × V)))) = ((^◡(1^st ↾ (V × V)) ∘ (𝐴 ∘ (1^st ↾ (V × V)))) ∩ (^◡(2^nd ↾ (V × V)) ∘ (𝐵 ∘ (2^nd ↾ (V × V)))))
3	1, 2	eqtri 2632	1 ⊢ pprod(𝐴, 𝐵) = ((^◡(1^st ↾ (V × V)) ∘ (𝐴 ∘ (1^st ↾ (V × V)))) ∩ (^◡(2^nd ↾ (V × V)) ∘ (𝐵 ∘ (2^nd ↾ (V × V)))))

Colors of variables: wff setvar class

Syntax hints: = wceq 1475 Vcvv 3173 ∩ cin 3539 × cxp 5036 ^◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 1^st c1st 7057 2^nd c2nd 7058 ⊗ ctxp 31106 pprodcpprod 31107

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-ext 2590

This theorem depends on definitions: df-bi 196 df-cleq 2603 df-txp 31130 df-pprod 31131

This theorem is referenced by: pprodcnveq 31160

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