hlass - Metamath Proof Explorer

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

Theorem hlass 31061

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

**Hypotheses**
Ref	Expression
hladdf.1	⊢ 𝑋 = (BaseSet‘𝑈)
hladdf.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
hlass	⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

**Proof of Theorem hlass**
Step	Hyp	Ref	Expression
1		hlnv 31051	. 2 ⊢ (𝑈 ∈ CHil_OLD → 𝑈 ∈ NrmCVec)
2		hladdf.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hladdf.2	. . 3 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
4	2, 3	nvass 30782	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))
5	1, 4	sylan 589	1 ⊢ ((𝑈 ∈ CHil_OLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = (𝐴𝐺(𝐵𝐺𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ‘cfv 6516 (class class class)co 7391 NrmCVeccnv 30744 +_𝑣 cpv 30745 BaseSetcba 30746 CHil_OLDchlo 31045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-ov 7394 df-oprab 7395 df-1st 7965 df-2nd 7966 df-grpo 30653 df-ablo 30705 df-vc 30719 df-nv 30752 df-va 30755 df-ba 30756 df-sm 30757 df-0v 30758 df-nmcv 30760 df-cbn 31023 df-hlo 31046

This theorem is referenced by: axhvass-zf 31144

W3C validator