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Theorem vcz 30662

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 30659	. . . . 5 ⊢ (𝑊 ∈ CVec_OLD → 𝑍 ∈ 𝑋)
5	4	anim2i 618	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑊 ∈ CVec_OLD) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋))
6	5	ancoms 458	. . 3 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋))
7		0cn 11136	. . . 4 ⊢ 0 ∈ ℂ
8		vc0.2	. . . . 5 ⊢ 𝑆 = (2^nd ‘𝑊)
9	1, 8, 2	vcass 30654	. . . 4 ⊢ ((𝑊 ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
10	7, 9	mp3anr2 1462	. . 3 ⊢ ((𝑊 ∈ CVec_OLD ∧ (𝐴 ∈ ℂ ∧ 𝑍 ∈ 𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
11	6, 10	syldan 592	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
12		mul01 11324	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
13	12	oveq1d 7383	. . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0)𝑆𝑍) = (0𝑆𝑍))
14	1, 8, 2, 3	vc0 30661	. . . 4 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝑍 ∈ 𝑋) → (0𝑆𝑍) = 𝑍)
15	4, 14	mpdan 688	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (0𝑆𝑍) = 𝑍)
16	13, 15	sylan9eqr 2794	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = 𝑍)
17	15	oveq2d 7384	. . 3 ⊢ (𝑊 ∈ CVec_OLD → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
18	17	adantr 480	. 2 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
19	11, 16, 18	3eqtr3rd 2781	1 ⊢ ((𝑊 ∈ CVec_OLD ∧ 𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ran crn 5633 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 ℂcc 11036 0cc0 11038 · cmul 11043 GIdcgi 30577 CVec_OLDcvc 30645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-1st 7943 df-2nd 7944 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-ltxr 11183 df-grpo 30580 df-gid 30581 df-ginv 30582 df-ablo 30632 df-vc 30646

This theorem is referenced by: nvsz 30725

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