hvmulcan - Hilbert Space Explorer

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Theorem hvmulcan 30834

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 2935	. . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2		biorf 933	. . . . 5 ⊢ (¬ 𝐴 = 0 → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
3	1, 2	sylbi 216	. . . 4 ⊢ (𝐴 ≠ 0 → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
4	3	ad2antlr 724	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
5	4	3adant3 1129	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
6		hvsubeq0 30830	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
7	6	3adant1 1127	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
8		hvsubdistr1 30811	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)))
9	8	eqeq1d 2728	. . . 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 30779	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		hvmul0or 30787	. . . . . 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 1112	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
14		hvmulcl 30775	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
15	14	3adant3 1129	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
16		hvmulcl 30775	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
17	16	3adant2 1128	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
18		hvsubeq0 30830	. . . . 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 1174	. 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 205 ∧ wa 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 (class class class)co 7405 ℂcc 11110 0cc0 11112 ℋchba 30681 ·_ℎ csm 30683 0_ℎc0v 30686 −_ℎ cmv 30687

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-hfvadd 30762 ax-hvcom 30763 ax-hvass 30764 ax-hv0cl 30765 ax-hvaddid 30766 ax-hfvmul 30767 ax-hvmulid 30768 ax-hvmulass 30769 ax-hvdistr1 30770 ax-hvdistr2 30771 ax-hvmul0 30772

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-hvsub 30733

This theorem is referenced by: hvsubcan 30836 hvsubcan2 30837

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