hlmulass - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hlmulass		Structured version Visualization version GIF version

Theorem hlmulass 28317

Description: Hilbert space scalar multiplication associative law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hlmulf.1	⊢ 𝑋 = (BaseSet‘𝑈)
hlmulf.4	⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)

**Assertion**
Ref	Expression
hlmulass	⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

**Proof of Theorem hlmulass**
Step	Hyp	Ref	Expression
1		hlnv 28302	. 2 ⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ NrmCVec)
2		hlmulf.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hlmulf.4	. . 3 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
4	2, 3	nvsass 28038	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))
5	1, 4	sylan 577	1 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ 𝑋)) → ((𝐴 · 𝐵)𝑆𝐶) = (𝐴𝑆(𝐵𝑆𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 ℂcc 10250 · cmul 10257 NrmCVeccnv 27994 BaseSetcba 27996 ·_𝑠OLD cns 27997 CHil_OLDchlo 28296

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-1st 7428 df-2nd 7429 df-vc 27969 df-nv 28002 df-va 28005 df-ba 28006 df-sm 28007 df-0v 28008 df-nmcv 28010 df-cbn 28274 df-hlo 28297

This theorem is referenced by: axhvmulass-zf 28401

W3C validator