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Theorem assaass 15113

Description: Left-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)

**Hypotheses**
Ref	Expression
isassa.v	⊢ 𝑉 = (Base‘𝑊)
isassa.f	⊢ 𝐹 = (Scalar‘𝑊)
isassa.b	⊢ 𝐵 = (Base‘𝐹)
isassa.s	⊢ · = ( ·_𝑠 ‘𝑊)
isassa.t	⊢ × = (._r‘𝑊)

**Assertion**
Ref	Expression
assaass	⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

**Proof of Theorem assaass**
Step	Hyp	Ref	Expression
1		isassa.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		isassa.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
3		isassa.b	. . 3 ⊢ 𝐵 = (Base‘𝐹)
4		isassa.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝑊)
5		isassa.t	. . 3 ⊢ × = (._r‘𝑊)
6	1, 2, 3, 4, 5	assalem 15112	. 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))))
7	6	simpld 112	1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ‘cfv 5380 (class class class)co 6088 Basecbs 13410 ._rcmulr 13491 Scalarcsca 13493 ·_𝑠 cvsca 13494 AssAlgcasa 15105

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-cnex 8276 ax-resscn 8277 ax-1re 8279 ax-addrcl 8282

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-iota 5340 df-fun 5382 df-fn 5383 df-fv 5388 df-ov 6091 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-5 9375 df-6 9376 df-ndx 13413 df-slot 13414 df-mulr 13504 df-sca 13506 df-vsca 13507 df-assa 15108

This theorem is used by: assa2ass 15118 assa2ass2 15119 issubassa3 15121 asclmul1 15138 assamulgscmlem2 15151

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