Theorem hvsubval 28206

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 6798	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝑦)))
2		oveq2 6799	. . 3 ⊢ (𝑦 = 𝐵 → (-1 ·ℎ 𝑦) = (-1 ·ℎ 𝐵))
3	2	oveq2d 6807	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ (-1 ·ℎ 𝑦)) = (𝐴 +ℎ (-1 ·ℎ 𝐵)))
4		df-hvsub 28161	. 2 ⊢ −ℎ = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑥 +ℎ (-1 ·ℎ 𝑦)))
5		ovex 6821	. 2 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐵)) ∈ V
6	1, 3, 4, 5	ovmpt2 6941	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  (class class class)co 6791  1c1 10137  -cneg 10467   ℋchil 28109   +_ℎ cva 28110   ·_ℎ csm 28111   −_ℎ cmv 28115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5992  df-fun 6031  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-hvsub 28161

This theorem is referenced by:  hvsubcl  28207  hvsubvali  28210  hvsubid  28216  hvnegid  28217  hv2neg  28218  hvaddsubval  28223  hvsub4  28227  hvaddsub12  28228  hvpncan  28229  hvaddsubass  28231  hvsubass  28234  hvsubdistr1  28239  hvsubdistr2  28240  hvsubcan  28264  hvsub0  28266  his2sub  28282  hhph  28368  shsubcl  28410  shsel3  28507  honegsubi  28988  lnopsubi  29166  lnfnsubi  29238  superpos  29546  cdj1i  29625