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Theorem ncvsi 25119

Description: The properties of a normed subcomplex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by AV, 7-Oct-2021.)

**Hypotheses**
Ref	Expression
isncvsngp.v	⊢ 𝑉 = (Base‘𝑊)
isncvsngp.n	⊢ 𝑁 = (norm‘𝑊)
isncvsngp.s	⊢ · = ( ·_𝑠 ‘𝑊)
isncvsngp.f	⊢ 𝐹 = (Scalar‘𝑊)
isncvsngp.k	⊢ 𝐾 = (Base‘𝐹)
ncvsi.m	⊢ − = (-_g‘𝑊)
ncvsi.0	⊢ 0 = (0_g‘𝑊)

**Assertion**
Ref	Expression
ncvsi	⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))

Distinct variable groups: 𝑘,𝐹,𝑥 𝑘,𝐾,𝑥 𝑘,𝑁,𝑥 𝑘,𝑉,𝑥 𝑘,𝑊,𝑥 · ,𝑘,𝑥 𝑦,𝑉 𝑦,𝑊,𝑥 Allowed substitution hints: · (𝑦) 𝐹(𝑦) 𝐾(𝑦) − (𝑥,𝑦,𝑘) 𝑁(𝑦) 0 (𝑥,𝑦,𝑘)

**Proof of Theorem ncvsi**
Step	Hyp	Ref	Expression
1		isncvsngp.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
2		isncvsngp.n	. . 3 ⊢ 𝑁 = (norm‘𝑊)
3		isncvsngp.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝑊)
4		isncvsngp.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
5		isncvsngp.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
6	1, 2, 3, 4, 5	isncvsngp 25117	. 2 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
7		simp1 1137	. . 3 ⊢ ((𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → 𝑊 ∈ ℂVec)
8	1, 2	nmf 24571	. . . 4 ⊢ (𝑊 ∈ NrmGrp → 𝑁:𝑉⟶ℝ)
9	8	3ad2ant2 1135	. . 3 ⊢ ((𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → 𝑁:𝑉⟶ℝ)
10		ncvsi.m	. . . . . . 7 ⊢ − = (-_g‘𝑊)
11		ncvsi.0	. . . . . . 7 ⊢ 0 = (0_g‘𝑊)
12	1, 2, 10, 11	ngpi 24584	. . . . . 6 ⊢ (𝑊 ∈ NrmGrp → (𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))
13		r19.26 3098	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) ↔ (∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
14		simpll 767	. . . . . . . . . . 11 ⊢ (((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
15		simplr 769	. . . . . . . . . . 11 ⊢ (((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
16		simpr 484	. . . . . . . . . . 11 ⊢ (((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))
17	14, 15, 16	3jca 1129	. . . . . . . . . 10 ⊢ (((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
18	17	ralimi 3075	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝑉 ((((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
19	13, 18	sylbir 235	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
20	19	ex 412	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) → (∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))
21	20	3ad2ant3 1136	. . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) → (∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))
22	12, 21	syl 17	. . . . 5 ⊢ (𝑊 ∈ NrmGrp → (∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))
23	22	imp 406	. . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
24	23	3adant1 1131	. . 3 ⊢ ((𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
25	7, 9, 24	3jca 1129	. 2 ⊢ ((𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))
26	6, 25	sylbi 217	1 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) → (𝑊 ∈ ℂVec ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)) ∧ ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∩ cin 3902 class class class wbr 5100 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ℝcr 11037 0cc0 11038 + caddc 11041 · cmul 11043 ≤ cle 11179 abscabs 15169 Basecbs 17148 Scalarcsca 17192 ·_𝑠 cvsca 17193 0_gc0g 17371 Grpcgrp 18875 -_gcsg 18877 normcnm 24532 NrmGrpcngp 24533 NrmVeccnvc 24537 ℂVecccvs 25091

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-cnex 11094 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 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

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-rmo 3352 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-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 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-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 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-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13279 df-fz 13436 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-rest 17354 df-topn 17355 df-0g 17373 df-topgen 17375 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20491 df-subrg 20515 df-abv 20754 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-xms 24276 df-ms 24277 df-nm 24538 df-ngp 24539 df-nrg 24541 df-nlm 24542 df-nvc 24543 df-clm 25031 df-cvs 25092

This theorem is referenced by: (None)

W3C validator