hvmulcan2 - Hilbert Space Explorer

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Theorem hvmulcan2 30314

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

**Assertion**
Ref	Expression
hvmulcan2	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 ·_ℎ 𝐶) = (𝐵 ·_ℎ 𝐶) ↔ 𝐴 = 𝐵))

**Proof of Theorem hvmulcan2**
Step	Hyp	Ref	Expression
1		hvmulcl 30254	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1132	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 30254	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1131	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubeq0 30309	. . . 4 ⊢ (((𝐴 ·_ℎ 𝐶) ∈ ℋ ∧ (𝐵 ·_ℎ 𝐶) ∈ ℋ) → (((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 ·_ℎ 𝐶) = (𝐵 ·_ℎ 𝐶)))
6	2, 4, 5	syl2anc 585	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 ·_ℎ 𝐶) = (𝐵 ·_ℎ 𝐶)))
7	6	3adant3r 1182	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → (((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ ↔ (𝐴 ·_ℎ 𝐶) = (𝐵 ·_ℎ 𝐶)))
8		hvsubdistr2 30291	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)))
9	8	eqeq1d 2735	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ))
10		subcl 11456	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
11		hvmul0or 30266	. . . . . 6 ⊢ (((𝐴 − 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
12	10, 11	stoic3 1779	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
13	9, 12	bitr3d 281	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
14	13	3adant3r 1182	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → (((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
15		df-ne 2942	. . . . . 6 ⊢ (𝐶 ≠ 0_ℎ ↔ ¬ 𝐶 = 0_ℎ)
16		biorf 936	. . . . . . 7 ⊢ (¬ 𝐶 = 0_ℎ → ((𝐴 − 𝐵) = 0 ↔ (𝐶 = 0_ℎ ∨ (𝐴 − 𝐵) = 0)))
17		orcom 869	. . . . . . 7 ⊢ ((𝐶 = 0_ℎ ∨ (𝐴 − 𝐵) = 0) ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ))
18	16, 17	bitrdi 287	. . . . . 6 ⊢ (¬ 𝐶 = 0_ℎ → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
19	15, 18	sylbi 216	. . . . 5 ⊢ (𝐶 ≠ 0_ℎ → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
20	19	ad2antll 728	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
21	20	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
22		subeq0 11483	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
23	22	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
24	14, 21, 23	3bitr2d 307	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → (((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ ↔ 𝐴 = 𝐵))
25	7, 24	bitr3d 281	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 ·_ℎ 𝐶) = (𝐵 ·_ℎ 𝐶) ↔ 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 (class class class)co 7406 ℂcc 11105 0cc0 11107 − cmin 11441 ℋchba 30160 ·_ℎ csm 30162 0_ℎc0v 30165 −_ℎ cmv 30166

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-hvcom 30242 ax-hvass 30243 ax-hv0cl 30244 ax-hvaddid 30245 ax-hfvmul 30246 ax-hvmulid 30247 ax-hvmulass 30248 ax-hvdistr2 30250 ax-hvmul0 30251

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-hvsub 30212

This theorem is referenced by: (None)

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