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Theorem vcz 27430

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 27427	. . . . 5 ⊢ (𝑊 ∈ CVec_OLD → 𝑍 ∈ 𝑋)
5	4	anim2i 593	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑊 ∈ CVec_OLD) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋))
6	5	ancoms 469	. . 3 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋))
7		0cn 10032	. . . 4 ⊢ 0 ∈ ℂ
8		vc0.2	. . . . 5 ⊢ 𝑆 = (2^nd ‘𝑊)
9	1, 8, 2	vcass 27422	. . . 4 ⊢ ((𝑊 ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
10	7, 9	mp3anr2 1422	. . 3 ⊢ ((𝑊 ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
11	6, 10	syldan 487	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
12		mul01 10215	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
13	12	oveq1d 6665	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0)𝑆𝑍) = (0𝑆𝑍))
14	1, 8, 2, 3	vc0 27429	. . . 4 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝑍 ∈ 𝑋) → (0𝑆𝑍) = 𝑍)
15	4, 14	mpdan 702	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (0𝑆𝑍) = 𝑍)
16	13, 15	sylan9eqr 2678	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = 𝑍)
17	15	oveq2d 6666	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
18	17	adantr 481	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
19	11, 16, 18	3eqtr3rd 2665	1 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ran crn 5115 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 ℂcc 9934 0cc0 9936 · cmul 9941 GIdcgi 27344 CVec_OLDcvc 27413

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-vc 27414

This theorem is referenced by: nvsz 27493

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