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Theorem ngppropd 22488
 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 22326 . . . . . . 7 (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
65adantr 480 . . . . . 6 ((𝜑𝐾 ∈ Grp) → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
71adantr 480 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐵 = (Base‘𝐾))
82adantr 480 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐵 = (Base‘𝐿))
9 simpr 476 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐾 ∈ Grp)
10 ngppropd.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1110adantlr 751 . . . . . . . . 9 (((𝜑𝐾 ∈ Grp) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
123adantr 480 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
137, 8, 9, 11, 12nmpropd2 22446 . . . . . . . 8 ((𝜑𝐾 ∈ Grp) → (norm‘𝐾) = (norm‘𝐿))
147, 8, 9, 11grpsubpropd2 17568 . . . . . . . 8 ((𝜑𝐾 ∈ Grp) → (-g𝐾) = (-g𝐿))
1513, 14coeq12d 5319 . . . . . . 7 ((𝜑𝐾 ∈ Grp) → ((norm‘𝐾) ∘ (-g𝐾)) = ((norm‘𝐿) ∘ (-g𝐿)))
161sqxpeqd 5175 . . . . . . . . . 10 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
1716reseq2d 5428 . . . . . . . . 9 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
182sqxpeqd 5175 . . . . . . . . . 10 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
1918reseq2d 5428 . . . . . . . . 9 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
203, 17, 193eqtr3d 2693 . . . . . . . 8 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
2120adantr 480 . . . . . . 7 ((𝜑𝐾 ∈ Grp) → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
2215, 21eqeq12d 2666 . . . . . 6 ((𝜑𝐾 ∈ Grp) → (((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↔ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
236, 22anbi12d 747 . . . . 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 674 . . . 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 17484 . . . . 5 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2625anbi1d 741 . . . 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 268 . . 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 1059 . . 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 1059 . . 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 303 . 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 2651 . . 3 (norm‘𝐾) = (norm‘𝐾)
32 eqid 2651 . . 3 (-g𝐾) = (-g𝐾)
33 eqid 2651 . . 3 (dist‘𝐾) = (dist‘𝐾)
34 eqid 2651 . . 3 (Base‘𝐾) = (Base‘𝐾)
35 eqid 2651 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
3631, 32, 33, 34, 35isngp2 22448 . 2 (𝐾 ∈ NrmGrp ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
37 eqid 2651 . . 3 (norm‘𝐿) = (norm‘𝐿)
38 eqid 2651 . . 3 (-g𝐿) = (-g𝐿)
39 eqid 2651 . . 3 (dist‘𝐿) = (dist‘𝐿)
40 eqid 2651 . . 3 (Base‘𝐿) = (Base‘𝐿)
41 eqid 2651 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
4237, 38, 39, 40, 41isngp2 22448 . 2 (𝐿 ∈ NrmGrp ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ MetSp ∧ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
4330, 36, 423bitr4g 303 1 (𝜑 → (𝐾 ∈ NrmGrp ↔ 𝐿 ∈ NrmGrp))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   × cxp 5141   ↾ cres 5145   ∘ ccom 5147  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  distcds 15997  TopOpenctopn 16129  Grpcgrp 17469  -gcsg 17471  MetSpcmt 22170  normcnm 22428  NrmGrpcngp 22429 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-0g 16149  df-topgen 16151  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-xms 22172  df-ms 22173  df-nm 22434  df-ngp 22435 This theorem is referenced by:  sranlm  22535
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