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Mirrors > Home > MPE Home > Th. List > ngpi | Structured version Visualization version GIF version |
Description: The properties of a normed group, which is a group accompanied by a norm. (Contributed by AV, 7-Oct-2021.) |
Ref | Expression |
---|---|
ngpi.v | ⊢ 𝑉 = (Base‘𝑊) |
ngpi.n | ⊢ 𝑁 = (norm‘𝑊) |
ngpi.m | ⊢ − = (-g‘𝑊) |
ngpi.0 | ⊢ 0 = (0g‘𝑊) |
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))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ngpgrp 23202 | . 2 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp) | |
2 | ngpi.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | ngpi.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
4 | 2, 3 | nmf 23218 | . 2 ⊢ (𝑊 ∈ NrmGrp → 𝑁:𝑉⟶ℝ) |
5 | ngpi.0 | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | 2, 3, 5 | nmeq0 23221 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) |
7 | ngpi.m | . . . . . . 7 ⊢ − = (-g‘𝑊) | |
8 | 2, 3, 7 | nmmtri 23225 | . . . . . 6 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
9 | 8 | 3expa 1114 | . . . . 5 ⊢ (((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) ∧ 𝑦 ∈ 𝑉) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
10 | 9 | ralrimiva 3182 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) |
11 | 6, 10 | jca 514 | . . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑥 ∈ 𝑉) → (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) |
12 | 11 | ralrimiva 3182 | . 2 ⊢ (𝑊 ∈ NrmGrp → ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))) |
13 | 1, 4, 12 | 3jca 1124 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 0cc0 10531 + caddc 10534 ≤ cle 10670 Basecbs 16477 0gc0g 16707 Grpcgrp 18097 -gcsg 18099 normcnm 23180 NrmGrpcngp 23181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 |
This theorem is referenced by: ncvsi 23749 |
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