Theorem nvaddsub 29246

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 29207	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
3		nvpncan2.1	. . . 4 ⊢ 𝑋 = (BaseSet‘𝑈)
4	3, 1	bafval 29195	. . 3 ⊢ 𝑋 = ran 𝐺
5		nvpncan2.3	. . . 4 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
6	1, 5	vsfval 29224	. . 3 ⊢ 𝑀 = ( /𝑔 ‘𝐺)
7	4, 6	ablomuldiv 29143	. 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 1540   ∈ wcel 2105  ‘cfv 6473  (class class class)co 7329  AbelOpcablo 29135  NrmCVeccnv 29175   +_𝑣 cpv 29176  BaseSetcba 29177   −_𝑣 cnsb 29180

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-1st 7891  df-2nd 7892  df-grpo 29084  df-gid 29085  df-ginv 29086  df-gdiv 29087  df-ablo 29136  df-vc 29150  df-nv 29183  df-va 29186  df-ba 29187  df-sm 29188  df-0v 29189  df-vs 29190  df-nmcv 29191

This theorem is referenced by:  nvnpcan  29247