Theorem nvaddsub 29017

Description: Commutative/associative law for vector addition and subtraction. (Contributed by NM, 24-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvpncan2.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvpncan2.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nvpncan2.3	⊢ 𝑀 = ( −𝑣 ‘𝑈)

Assertion

Ref	Expression
nvaddsub	⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵))

Proof of Theorem nvaddsub

Step	Hyp	Ref	Expression
1		nvpncan2.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2	1	nvablo 28978	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
3		nvpncan2.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 28966	. . 3 ⊢ 𝑋 = ran 𝐺
5		nvpncan2.3	. . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
6	1, 5	vsfval 28995	. . 3 ⊢ 𝑀 = ( /𝑔 ‘𝐺)
7	4, 6	ablomuldiv 28914	. 2 ⊢ ((𝐺 ∈ AbelOp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵))
8	2, 7	sylan 580	1 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺𝐵)𝑀𝐶) = ((𝐴𝑀𝐶)𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  AbelOpcablo 28906  NrmCVeccnv 28946   +_𝑣 cpv 28947  BaseSetcba 28948   −_𝑣 cnsb 28951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ginv 28857  df-gdiv 28858  df-ablo 28907  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-vs 28961  df-nmcv 28962

This theorem is referenced by:  nvnpcan  29018