hvmulcan - Hilbert Space Explorer

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Theorem hvmulcan 31231

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 2957	. . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0)
2		biorf 947	. . . . 5 ⊢ (¬ 𝐴 = 0 → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
3	1, 2	sylbi 219	. . . 4 ⊢ (𝐴 ≠ 0 → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
4	3	ad2antlr 737	. . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
5	4	3adant3 1144	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
6		hvsubeq0 31227	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
7	6	3adant1 1142	. 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 −_ℎ 𝐶) = 0_ℎ ↔ 𝐵 = 𝐶))
8		hvsubdistr1 31208	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = ((𝐴 ·_ℎ 𝐵) −_ℎ (𝐴 ·_ℎ 𝐶)))
9	8	eqeq1d 2763	. . . 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 31176	. . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −_ℎ 𝐶) ∈ ℋ)
11		hvmul0or 31184	. . . . . 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 602	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
13	12	3impb 1126	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·_ℎ (𝐵 −_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ)))
14		hvmulcl 31172	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
15	14	3adant3 1144	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐵) ∈ ℋ)
16		hvmulcl 31172	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
17	16	3adant2 1143	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
18		hvsubeq0 31227	. . . . 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 593	. . . 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 311	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 = 0 ∨ (𝐵 −_ℎ 𝐶) = 0_ℎ) ↔ (𝐴 ·_ℎ 𝐵) = (𝐴 ·_ℎ 𝐶)))
21	20	3adant1r 1190	. 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 312	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 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 (class class class)co 7390 ℂcc 11064 0cc0 11066 ℋchba 31078 ·_ℎ csm 31080 0_ℎc0v 31083 −_ℎ cmv 31084

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-hfvadd 31159 ax-hvcom 31160 ax-hvass 31161 ax-hv0cl 31162 ax-hvaddid 31163 ax-hfvmul 31164 ax-hvmulid 31165 ax-hvmulass 31166 ax-hvdistr1 31167 ax-hvdistr2 31168 ax-hvmul0 31169

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-po 5551 df-so 5552 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8671 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-hvsub 31130

This theorem is referenced by: hvsubcan 31233 hvsubcan2 31234

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