Theorem nvm 30801

Description: Vector subtraction in terms of group division operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvm.1	⊢ 𝑋 = (BaseSet‘𝑈)
nvm.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nvm.3	⊢ 𝑀 = ( −𝑣 ‘𝑈)
nvm.6	⊢ 𝑁 = ( /𝑔 ‘𝐺)

Assertion

Ref	Expression
nvm	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))

Proof of Theorem nvm

Step	Hyp	Ref	Expression
1		nvm.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2		nvm.3	. . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
3	1, 2	vsfval 30793	. . . 4 ⊢ 𝑀 = ( /𝑔 ‘𝐺)
4		nvm.6	. . . 4 ⊢ 𝑁 = ( /𝑔 ‘𝐺)
5	3, 4	eqtr4i 2787	. . 3 ⊢ 𝑀 = 𝑁
6	5	oveqi 7404	. 2 ⊢ (𝐴𝑀𝐵) = (𝐴𝑁𝐵)
7	6	a1i 11	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑀𝐵) = (𝐴𝑁𝐵))

Syntax hints:   → wi 4   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ‘cfv 6516  (class class class)co 7391   /_𝑔 cgs 30652  NrmCVeccnv 30744   +_𝑣 cpv 30745  BaseSetcba 30746   −_𝑣 cnsb 30749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-grpo 30653  df-gdiv 30656  df-va 30755  df-vs 30759

This theorem is referenced by:  nvmval  30802