hvaddsub12 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > hvaddsub12		Structured version Visualization version GIF version

Theorem hvaddsub12 30974

Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hvaddsub12	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)))

**Proof of Theorem hvaddsub12**
Step	Hyp	Ref	Expression
1		neg1cn 12178	. . . 4 ⊢ -1 ∈ ℂ
2		hvmulcl 30949	. . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·_ℎ 𝐶) ∈ ℋ)
3	1, 2	mpan 690	. . 3 ⊢ (𝐶 ∈ ℋ → (-1 ·_ℎ 𝐶) ∈ ℋ)
4		hvadd12 30971	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (-1 ·_ℎ 𝐶) ∈ ℋ) → (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
5	3, 4	syl3an3 1165	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
6		hvsubval 30952	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) = (𝐵 +_ℎ (-1 ·_ℎ 𝐶)))
7	6	oveq2d 7406	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
8	7	3adant1 1130	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐴 +_ℎ (𝐵 +_ℎ (-1 ·_ℎ 𝐶))))
9		hvsubval 30952	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −_ℎ 𝐶) = (𝐴 +_ℎ (-1 ·_ℎ 𝐶)))
10	9	oveq2d 7406	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
11	10	3adant2 1131	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 +_ℎ (-1 ·_ℎ 𝐶))))
12	5, 8, 11	3eqtr4d 2775	1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +_ℎ (𝐵 −_ℎ 𝐶)) = (𝐵 +_ℎ (𝐴 −_ℎ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 (class class class)co 7390 ℂcc 11073 1c1 11076 -cneg 11413 ℋchba 30855 +_ℎ cva 30856 ·_ℎ csm 30857 −_ℎ cmv 30861

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-hvcom 30937 ax-hvass 30938 ax-hfvmul 30941

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-ltxr 11220 df-sub 11414 df-neg 11415 df-hvsub 30907

This theorem is referenced by: 5oalem1 31590 3oalem2 31599 pjcji 31620 pjclem4 32135 pj3si 32143

W3C validator