nvadd32 - Metamath Proof Explorer

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Theorem nvadd32 28384

Description: Commutative/associative law for vector addition. (Contributed by NM, 27-Dec-2007.) (New usage is discouraged.)

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

**Assertion**
Ref	Expression
nvadd32	⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

**Proof of Theorem nvadd32**
Step	Hyp	Ref	Expression
1		nvgcl.2	. . 3 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
2	1	nvablo 28377	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
3		nvgcl.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28365	. . 3 ⊢ 𝑋 = ran 𝐺
5	4	ablo32 28310	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))
6	2, 5	sylan 582	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐺𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 AbelOpcablo 28305 NrmCVeccnv 28345 +_𝑣 cpv 28346 BaseSetcba 28347

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-1st 7675 df-2nd 7676 df-grpo 28254 df-ablo 28306 df-vc 28320 df-nv 28353 df-va 28356 df-ba 28357 df-sm 28358 df-0v 28359 df-nmcv 28361

This theorem is referenced by: nvpncan2 28414

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