Theorem hvsubval 31648

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 7450	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ ( -1 ·ℎ 𝑦)) = (𝐴 +ℎ ( -1 ·ℎ 𝑦)))
2		oveq2 7451	. . 3 ⊢ (𝑦 = 𝐵 → ( -1 ·ℎ 𝑦) = ( -1 ·ℎ 𝐵))
3	2	oveq2d 7459	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ ( -1 ·ℎ 𝑦)) = (𝐴 +ℎ ( -1 ·ℎ 𝐵)))
4		df-hvsub 31603	. 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ ( -1 ·ℎ 𝑦)))
5		ovex 7476	. 2 ⊢ (𝐴 +ℎ ( -1 ·ℎ 𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7604	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ ( -1 ·ℎ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  (class class class)co 7443  1c1 11228   -cneg 11569   ℋchba 31551   +_ℎ cva 31552   ·_ℎ csm 31553   −_ℎ cmv 31557

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-hvsub 31603

This theorem is used by:  hvsubcl  31649  hvsubvali  31652  hvsubid  31658  hvnegid  31659  hv2neg  31660  hvaddsubval  31665  hvsub4  31669  hvaddsub12  31670  hvpncan  31671  hvaddsubass  31673  hvsubass  31676  hvsubdistr1  31681  hvsubdistr2  31682  hvsubcan  31706  hvsub0  31708  his2sub  31724  hhph  31810  shsubcl  31852  shsel3  31947  honegsubi  32428  lnopsubi  32606  lnfnsubi  32678  superpos  32986  cdj1i  33065