Theorem hvsubval 31091

Description: Value of vector subtraction. (Contributed by NM, 5-Sep-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
hvsubval	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)))

Proof of Theorem hvsubval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7365	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝑦)))
2		oveq2 7366	. . 3 ⊢ (𝑦 = 𝐵 → (-1 ·ℎ 𝑦) = (-1 ·ℎ 𝐵))
3	2	oveq2d 7374	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝐵)))
4		df-hvsub 31046	. 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦)))
5		ovex 7391	. 2 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7518	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  (class class class)co 7358  1c1 11027  -cneg 11365   ℋchba 30994   +_ℎ cva 30995   ·_ℎ csm 30996   −_ℎ cmv 31000

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-hvsub 31046

This theorem is referenced by:  hvsubcl  31092  hvsubvali  31095  hvsubid  31101  hvnegid  31102  hv2neg  31103  hvaddsubval  31108  hvsub4  31112  hvaddsub12  31113  hvpncan  31114  hvaddsubass  31116  hvsubass  31119  hvsubdistr1  31124  hvsubdistr2  31125  hvsubcan  31149  hvsub0  31151  his2sub  31167  hhph  31253  shsubcl  31295  shsel3  31390  honegsubi  31871  lnopsubi  32049  lnfnsubi  32121  superpos  32429  cdj1i  32508