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Theorem vcz 27766

Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
vc0.1	⊢ 𝐺 = (1^st ‘𝑊)
vc0.2	⊢ 𝑆 = (2^nd ‘𝑊)
vc0.3	⊢ 𝑋 = ran 𝐺
vc0.4	⊢ 𝑍 = (GId‘𝐺)

**Assertion**
Ref	Expression
vcz	⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

**Proof of Theorem vcz**
Step	Hyp	Ref	Expression
1		vc0.1	. . . . . 6 ⊢ 𝐺 = (1^st ‘𝑊)
2		vc0.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
3		vc0.4	. . . . . 6 ⊢ 𝑍 = (GId‘𝐺)
4	1, 2, 3	vczcl 27763	. . . . 5 ⊢ (𝑊 ∈ CVec_OLD → 𝑍 ∈ 𝑋)
5	4	anim2i 603	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑊 ∈ CVec_OLD) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋))
6	5	ancoms 446	. . 3 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋))
7		0cn 10234	. . . 4 ⊢ 0 ∈ ℂ
8		vc0.2	. . . . 5 ⊢ 𝑆 = (2^nd ‘𝑊)
9	1, 8, 2	vcass 27758	. . . 4 ⊢ ((𝑊 ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
10	7, 9	mp3anr2 1570	. . 3 ⊢ ((𝑊 ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
11	6, 10	syldan 579	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
12		mul01 10417	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
13	12	oveq1d 6807	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0)𝑆𝑍) = (0𝑆𝑍))
14	1, 8, 2, 3	vc0 27765	. . . 4 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝑍 ∈ 𝑋) → (0𝑆𝑍) = 𝑍)
15	4, 14	mpdan 667	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (0𝑆𝑍) = 𝑍)
16	13, 15	sylan9eqr 2827	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = 𝑍)
17	15	oveq2d 6808	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
18	17	adantr 466	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
19	11, 16, 18	3eqtr3rd 2814	1 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ran crn 5250 ‘cfv 6029 (class class class)co 6792 1^st c1st 7313 2^nd c2nd 7314 ℂcc 10136 0cc0 10138 · cmul 10143 GIdcgi 27680 CVec_OLDcvc 27749

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-1st 7315 df-2nd 7316 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 df-grpo 27683 df-gid 27684 df-ginv 27685 df-ablo 27735 df-vc 27750

This theorem is referenced by: nvsz 27829

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