hvmulcan - Hilbert Space Explorer

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Theorem hvmulcan 31147

Description: Cancellation law for scalar multiplication. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hvmulcan	⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶) ↔ 𝐵 = 𝐶))

**Proof of Theorem hvmulcan**
Step	Hyp	Ref	Expression
1		df-ne 2933	. . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2		biorf 936	. . . . 5 ⊢ (¬ 𝐴 = 0 → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
3	1, 2	sylbi 217	. . . 4 ⊢ (𝐴 ≠ 0 → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
4	3	ad2antlr 727	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
5	4	3adant3 1132	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
6		hvsubeq0 31143	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
7	6	3adant1 1130	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
8		hvsubdistr1 31124	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)))
9	8	eqeq1d 2738	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)) = 0_ℎ))
10		hvsubcl 31092	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		hvmul0or 31100	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 −_ℎ 𝐶) ∈ ℋ) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
12	10, 11	sylan2 593	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
13	12	3impb 1114	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
14		hvmulcl 31088	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
15	14	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
16		hvmulcl 31088	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
17	16	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
18		hvsubeq0 31143	. . . . 5 ⊢ (((𝐴 ·_ℎ 𝐵) ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶)))
19	15, 17, 18	syl2anc 584	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶)))
20	9, 13, 19	3bitr3d 309	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ) ↔ (𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶)))
21	20	3adant1r 1178	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ) ↔ (𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶)))
22	5, 7, 21	3bitr3rd 310	1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶) ↔ 𝐵 = 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 (class class class)co 7358 ℂcc 11024 0cc0 11026 ℋchba 30994 ·_ℎ csm 30996 0_ℎc0v 30999 −_ℎ cmv 31000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvmulass 31082 ax-hvdistr1 31083 ax-hvdistr2 31084 ax-hvmul0 31085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-hvsub 31046

This theorem is referenced by: hvsubcan 31149 hvsubcan2 31150

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