hvmulcan2 - Hilbert Space Explorer

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Theorem hvmulcan2 30761

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 30701	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1128	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 30701	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1127	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubeq0 30756	. . . 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 583	. . 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 1178	. 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 30738	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)))
9	8	eqeq1d 2726	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ))
10		subcl 11455	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
11		hvmul0or 30713	. . . . . 6 ⊢ (((𝐴 − 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
12	10, 11	stoic3 1770	. . . . 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 1178	. . 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 2933	. . . . . 6 ⊢ (𝐶 ≠ 0_ℎ ↔ ¬ 𝐶 = 0_ℎ)
16		biorf 933	. . . . . . 7 ⊢ (¬ 𝐶 = 0_ℎ → ((𝐴 − 𝐵) = 0 ↔ (𝐶 = 0_ℎ ∨ (𝐴 − 𝐵) = 0)))
17		orcom 867	. . . . . . 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 726	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
21	20	3adant1 1127	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
22		subeq0 11482	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
23	22	3adant3 1129	. . 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 395 ∨ wo 844 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 (class class class)co 7401 ℂcc 11103 0cc0 11105 − cmin 11440 ℋchba 30607 ·_ℎ csm 30609 0_ℎc0v 30612 −_ℎ cmv 30613

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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-hvcom 30689 ax-hvass 30690 ax-hv0cl 30691 ax-hvaddid 30692 ax-hfvmul 30693 ax-hvmulid 30694 ax-hvmulass 30695 ax-hvdistr2 30697 ax-hvmul0 30698

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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-hvsub 30659

This theorem is referenced by: (None)

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