hvmulcan2 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > hvmulcan2		Structured version Visualization version GIF version

Theorem hvmulcan2 29414

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 29354	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2	1	3adant2 1129	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
3		hvmulcl 29354	. . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
4	3	3adant1 1128	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐵 ·_ℎ 𝐶) ∈ ℋ)
5		hvsubeq0 29409	. . . 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 1179	. 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 29391	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → ((𝐴 − 𝐵) ·_ℎ 𝐶) = ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)))
9	8	eqeq1d 2741	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 ·_ℎ 𝐶) −_ℎ (𝐵 ·_ℎ 𝐶)) = 0_ℎ))
10		subcl 11203	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
11		hvmul0or 29366	. . . . . 6 ⊢ (((𝐴 − 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
12	10, 11	stoic3 1782	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (((𝐴 − 𝐵) ·_ℎ 𝐶) = 0_ℎ ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
13	9, 12	bitr3d 280	. . . 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 1179	. . 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 2945	. . . . . 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 866	. . . . . . 7 ⊢ ((𝐶 = 0_ℎ ∨ (𝐴 − 𝐵) = 0) ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ))
18	16, 17	bitrdi 286	. . . . . 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 725	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
21	20	3adant1 1128	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ ((𝐴 − 𝐵) = 0 ∨ 𝐶 = 0_ℎ)))
22		subeq0 11230	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
23	22	3adant3 1130	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℋ ∧ 𝐶 ≠ 0_ℎ)) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵))
24	14, 21, 23	3bitr2d 306	. 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 280	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 843 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 (class class class)co 7268 ℂcc 10853 0cc0 10855 − cmin 11188 ℋchba 29260 ·_ℎ csm 29262 0_ℎc0v 29265 −_ℎ cmv 29266

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-hvcom 29342 ax-hvass 29343 ax-hv0cl 29344 ax-hvaddid 29345 ax-hfvmul 29346 ax-hvmulid 29347 ax-hvmulass 29348 ax-hvdistr2 29350 ax-hvmul0 29351

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-po 5502 df-so 5503 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-hvsub 29312

This theorem is referenced by: (None)

W3C validator