Theorem hvsubval 31623

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 7427	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ ( -1 ·ℎ 𝑦)) = (𝐴 +ℎ ( -1 ·ℎ 𝑦)))
2		oveq2 7428	. . 3 ⊢ (𝑦 = 𝐵 → ( -1 ·ℎ 𝑦) = ( -1 ·ℎ 𝐵))
3	2	oveq2d 7436	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ ( -1 ·ℎ 𝑦)) = (𝐴 +ℎ ( -1 ·ℎ 𝐵)))
4		df-hvsub 31578	. 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ ( -1 ·ℎ 𝑦)))
5		ovex 7453	. 2 ⊢ (𝐴 +ℎ ( -1 ·ℎ 𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7580	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ ( -1 ·ℎ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  (class class class)co 7420  1c1 11204   -cneg 11545   ℋchba 31526   +_ℎ cva 31527   ·_ℎ csm 31528   −_ℎ cmv 31532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-hvsub 31578

This theorem is used by:  hvsubcl  31624  hvsubvali  31627  hvsubid  31633  hvnegid  31634  hv2neg  31635  hvaddsubval  31640  hvsub4  31644  hvaddsub12  31645  hvpncan  31646  hvaddsubass  31648  hvsubass  31651  hvsubdistr1  31656  hvsubdistr2  31657  hvsubcan  31681  hvsub0  31683  his2sub  31699  hhph  31785  shsubcl  31827  shsel3  31922  honegsubi  32403  lnopsubi  32581  lnfnsubi  32653  superpos  32961  cdj1i  33040