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Theorem ngppropd 23173
Description: Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
ngppropd.1 (𝜑𝐵 = (Base‘𝐾))
ngppropd.2 (𝜑𝐵 = (Base‘𝐿))
ngppropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
ngppropd.4 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
ngppropd.5 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
Assertion
Ref Expression
ngppropd (𝜑 → (𝐾 ∈ NrmGrp ↔ 𝐿 ∈ NrmGrp))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem ngppropd
StepHypRef Expression
1 ngppropd.1 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐾))
2 ngppropd.2 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐿))
3 ngppropd.4 . . . . . . . 8 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
4 ngppropd.5 . . . . . . . 8 (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
51, 2, 3, 4mspropd 23011 . . . . . . 7 (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
65adantr 481 . . . . . 6 ((𝜑𝐾 ∈ Grp) → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
71adantr 481 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐵 = (Base‘𝐾))
82adantr 481 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐵 = (Base‘𝐿))
9 simpr 485 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐾 ∈ Grp)
10 ngppropd.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1110adantlr 711 . . . . . . . . 9 (((𝜑𝐾 ∈ Grp) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
123adantr 481 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
137, 8, 9, 11, 12nmpropd2 23131 . . . . . . . 8 ((𝜑𝐾 ∈ Grp) → (norm‘𝐾) = (norm‘𝐿))
147, 8, 9, 11grpsubpropd2 18143 . . . . . . . 8 ((𝜑𝐾 ∈ Grp) → (-g𝐾) = (-g𝐿))
1513, 14coeq12d 5728 . . . . . . 7 ((𝜑𝐾 ∈ Grp) → ((norm‘𝐾) ∘ (-g𝐾)) = ((norm‘𝐿) ∘ (-g𝐿)))
161sqxpeqd 5580 . . . . . . . . . 10 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
1716reseq2d 5846 . . . . . . . . 9 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
182sqxpeqd 5580 . . . . . . . . . 10 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
1918reseq2d 5846 . . . . . . . . 9 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
203, 17, 193eqtr3d 2861 . . . . . . . 8 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
2120adantr 481 . . . . . . 7 ((𝜑𝐾 ∈ Grp) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
2215, 21eqeq12d 2834 . . . . . 6 ((𝜑𝐾 ∈ Grp) → (((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↔ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
236, 22anbi12d 630 . . . . 5 ((𝜑𝐾 ∈ Grp) → ((𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
2423pm5.32da 579 . . . 4 (𝜑 → ((𝐾 ∈ Grp ∧ (𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) ↔ (𝐾 ∈ Grp ∧ (𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))))
251, 2, 10grppropd 18056 . . . . 5 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2625anbi1d 629 . . . 4 (𝜑 → ((𝐾 ∈ Grp ∧ (𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))) ↔ (𝐿 ∈ Grp ∧ (𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))))
2724, 26bitrd 280 . . 3 (𝜑 → ((𝐾 ∈ Grp ∧ (𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))) ↔ (𝐿 ∈ Grp ∧ (𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))))
28 3anass 1087 . . 3 ((𝐾 ∈ Grp ∧ 𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (𝐾 ∈ Grp ∧ (𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))))
29 3anass 1087 . . 3 ((𝐿 ∈ Grp ∧ 𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))) ↔ (𝐿 ∈ Grp ∧ (𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
3027, 28, 293bitr4g 315 . 2 (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))))
31 eqid 2818 . . 3 (norm‘𝐾) = (norm‘𝐾)
32 eqid 2818 . . 3 (-g𝐾) = (-g𝐾)
33 eqid 2818 . . 3 (dist‘𝐾) = (dist‘𝐾)
34 eqid 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
35 eqid 2818 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
3631, 32, 33, 34, 35isngp2 23133 . 2 (𝐾 ∈ NrmGrp ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
37 eqid 2818 . . 3 (norm‘𝐿) = (norm‘𝐿)
38 eqid 2818 . . 3 (-g𝐿) = (-g𝐿)
39 eqid 2818 . . 3 (dist‘𝐿) = (dist‘𝐿)
40 eqid 2818 . . 3 (Base‘𝐿) = (Base‘𝐿)
41 eqid 2818 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
4237, 38, 39, 40, 41isngp2 23133 . 2 (𝐿 ∈ NrmGrp ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
4330, 36, 423bitr4g 315 1 (𝜑 → (𝐾 ∈ NrmGrp ↔ 𝐿 ∈ NrmGrp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   × cxp 5546  cres 5550  ccom 5552  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  distcds 16562  TopOpenctopn 16683  Grpcgrp 18041  -gcsg 18043  MetSpcms 22855  normcnm 23113  NrmGrpcngp 23114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-0g 16703  df-topgen 16705  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044  df-minusg 18045  df-sbg 18046  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-xms 22857  df-ms 22858  df-nm 23119  df-ngp 23120
This theorem is referenced by:  sranlm  23220
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