hvmulcan - Hilbert Space Explorer

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Theorem hvmulcan 31104

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 2947	. . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2		biorf 935	. . . . 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 726	. . 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 31100	. . 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 31081	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)))
9	8	eqeq1d 2742	. . . 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 31049	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		hvmul0or 31057	. . . . . 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 592	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
13	12	3impb 1115	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
14		hvmulcl 31045	. . . . . 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 31045	. . . . . 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 31100	. . . . 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 583	. . . 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 1177	. 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 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 (class class class)co 7448 ℂcc 11182 0cc0 11184 ℋchba 30951 ·_ℎ csm 30953 0_ℎc0v 30956 −_ℎ cmv 30957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvmulass 31039 ax-hvdistr1 31040 ax-hvdistr2 31041 ax-hvmul0 31042

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-hvsub 31003

This theorem is referenced by: hvsubcan 31106 hvsubcan2 31107

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