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Theorem ngpi 24589

Description: The properties of a normed group, which is a group accompanied by a norm. (Contributed by AV, 7-Oct-2021.)

**Hypotheses**
Ref	Expression
ngpi.v	⊢ 𝑉 = (Base‘𝑊)
ngpi.n	⊢ 𝑁 = (norm‘𝑊)
ngpi.m	⊢ − = (-_g‘𝑊)
ngpi.0	⊢ 0 = (0_g‘𝑊)

**Assertion**
Ref	Expression
ngpi	⊢ (𝑊 ∈ NrmGrp → (𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))

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

**Proof of Theorem ngpi**
Step	Hyp	Ref	Expression
1		ngpgrp 24560	. 2 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp)
2		ngpi.v	. . 3 ⊢ 𝑉 = (Base‘𝑊)
3		ngpi.n	. . 3 ⊢ 𝑁 = (norm‘𝑊)
4	2, 3	nmf 24576	. 2 ⊢ (𝑊 ∈ NrmGrp → 𝑁:𝑉⟶ℝ)
5		ngpi.0	. . . . 5 ⊢ 0 = (0_g‘𝑊)
6	2, 3, 5	nmeq0 24579	. . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ))
7		ngpi.m	. . . . . . 7 ⊢ − = (-_g‘𝑊)
8	2, 3, 7	nmmtri 24583	. . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
9	8	3expa 1119	. . . . 5 ⊢ (((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
10	9	ralrimiva 3130	. . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))
11	6, 10	jca 511	. . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
12	11	ralrimiva 3130	. 2 ⊢ (𝑊 ∈ NrmGrp → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))
13	1, 4, 12	3jca 1129	1 ⊢ (𝑊 ∈ NrmGrp → (𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 class class class wbr 5100 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ℝcr 11039 0cc0 11040 + caddc 11043 ≤ cle 11181 Basecbs 17150 0_gc0g 17373 Grpcgrp 18880 -_gcsg 18882 normcnm 24537 NrmGrpcngp 24538

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-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 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-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-er 8647 df-map 8779 df-en 8898 df-dom 8899 df-sdom 8900 df-sup 9359 df-inf 9360 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-n0 12416 df-z 12503 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-0g 17375 df-topgen 17377 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-grp 18883 df-minusg 18884 df-sbg 18885 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-top 22855 df-topon 22872 df-topsp 22894 df-bases 22907 df-xms 24281 df-ms 24282 df-nm 24543 df-ngp 24544

This theorem is referenced by: ncvsi 25124

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