hvmulcan - Hilbert Space Explorer

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Theorem hvmulcan 31159

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 2934	. . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2		biorf 937	. . . . 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 728	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
5	4	3adant3 1133	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
6		hvsubeq0 31155	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
7	6	3adant1 1131	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
8		hvsubdistr1 31136	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)))
9	8	eqeq1d 2739	. . . 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 31104	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		hvmul0or 31112	. . . . . 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 594	. . . . 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 31100	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
15	14	3adant3 1133	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
16		hvmulcl 31100	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
17	16	3adant2 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
18		hvsubeq0 31155	. . . . 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 585	. . . 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 1179	. 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 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 (class class class)co 7368 ℂcc 11036 0cc0 11038 ℋchba 31006 ·_ℎ csm 31008 0_ℎc0v 31011 −_ℎ cmv 31012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-hfvadd 31087 ax-hvcom 31088 ax-hvass 31089 ax-hv0cl 31090 ax-hvaddid 31091 ax-hfvmul 31092 ax-hvmulid 31093 ax-hvmulass 31094 ax-hvdistr1 31095 ax-hvdistr2 31096 ax-hvmul0 31097

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-hvsub 31058

This theorem is referenced by: hvsubcan 31161 hvsubcan2 31162

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