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Theorem ncvsi 25347

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 25345	. 2 ⊢ (𝑊 ∈ (NrmVec ∩ ℂVec) ↔ (𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))))
7		simp1 1154	. . 3 ⊢ ((𝑊 ∈ ℂVec ∧ 𝑊 ∈ NrmGrp ∧ ∀𝑥 ∈ 𝑉 ∀𝑘 ∈ 𝐾 (𝑁‘(𝑘 · 𝑥)) = ((abs‘𝑘) · (𝑁‘𝑥))) → 𝑊 ∈ ℂVec)
8	1, 2	nmf 24809	. . . 4 ⊢ (𝑊 ∈ NrmGrp → 𝑁:𝑉⟶ℝ)
9	8	3ad2ant2 1152	. . 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 24822	. . . . . 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 3128	. . . . . . . . 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 779	. . . . . . . . . . 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 781	. . . . . . . . . . 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 490	. . . . . . . . . . 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 1146	. . . . . . . . . 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 3105	. . . . . . . . 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 238	. . . . . . . 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 418	. . . . . . 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 1153	. . . . . 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 18	. . . . 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 412	. . . 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 1148	. . 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 1146	. 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 220	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3082 ∩ cin 3907 class class class wbr 5114 ⟶wf 6539 ‘cfv 6543 (class class class)co 7423 ℝcr 11117 0cc0 11118 + caddc 11121 · cmul 11123 ≤ cle 11262 abscabs 15311 Basecbs 17294 Scalarcsca 17338 ·_𝑠 cvsca 17339 0_gc0g 17517 Grpcgrp 19031 -_gcsg 19033 normcnm 24770 NrmGrpcngp 24771 NrmVeccnvc 24775 ℂVecccvs 25319

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197 ax-mulf 11198

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13396 df-fz 13554 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-rest 17500 df-topn 17501 df-0g 17519 df-topgen 17521 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-grp 19034 df-minusg 19035 df-sbg 19036 df-subg 19220 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-ring 20348 df-cring 20349 df-subrng 20682 df-subrg 20706 df-abv 20949 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-cnfld 21560 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-xms 24514 df-ms 24515 df-nm 24776 df-ngp 24777 df-nrg 24779 df-nlm 24780 df-nvc 24781 df-clm 25259 df-cvs 25320

This theorem is used by: (None)

W3C validator