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Theorem ngppropd 22354
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 22192 . . . . . . 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 477 . . . . . . . . 9 ((𝜑𝐾 ∈ Grp) → 𝐾 ∈ Grp)
10 ngppropd.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1110adantlr 750 . . . . . . . . 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 22312 . . . . . . . 8 ((𝜑𝐾 ∈ Grp) → (norm‘𝐾) = (norm‘𝐿))
147, 8, 9, 11grpsubpropd2 17445 . . . . . . . 8 ((𝜑𝐾 ∈ Grp) → (-g𝐾) = (-g𝐿))
1513, 14coeq12d 5248 . . . . . . 7 ((𝜑𝐾 ∈ Grp) → ((norm‘𝐾) ∘ (-g𝐾)) = ((norm‘𝐿) ∘ (-g𝐿)))
161sqxpeqd 5103 . . . . . . . . . 10 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
1716reseq2d 5358 . . . . . . . . 9 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
182sqxpeqd 5103 . . . . . . . . . 10 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
1918reseq2d 5358 . . . . . . . . 9 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
203, 17, 193eqtr3d 2663 . . . . . . . 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 2636 . . . . . 6 ((𝜑𝐾 ∈ Grp) → (((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) ↔ ((norm‘𝐿) ∘ (-g𝐿)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))))
236, 22anbi12d 746 . . . . 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 672 . . . 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 17361 . . . . 5 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2625anbi1d 740 . . . 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 1040 . . 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 1040 . . 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 2621 . . 3 (norm‘𝐾) = (norm‘𝐾)
32 eqid 2621 . . 3 (-g𝐾) = (-g𝐾)
33 eqid 2621 . . 3 (dist‘𝐾) = (dist‘𝐾)
34 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
35 eqid 2621 . . 3 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
3631, 32, 33, 34, 35isngp2 22314 . 2 (𝐾 ∈ NrmGrp ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ MetSp ∧ ((norm‘𝐾) ∘ (-g𝐾)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))))
37 eqid 2621 . . 3 (norm‘𝐿) = (norm‘𝐿)
38 eqid 2621 . . 3 (-g𝐿) = (-g𝐿)
39 eqid 2621 . . 3 (dist‘𝐿) = (dist‘𝐿)
40 eqid 2621 . . 3 (Base‘𝐿) = (Base‘𝐿)
41 eqid 2621 . . 3 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
4237, 38, 39, 40, 41isngp2 22314 . 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 384  w3a 1036   = wceq 1480  wcel 1987   × cxp 5074  cres 5078  ccom 5080  cfv 5849  (class class class)co 6607  Basecbs 15784  +gcplusg 15865  distcds 15874  TopOpenctopn 16006  Grpcgrp 17346  -gcsg 17348  MetSpcmt 22036  normcnm 22294  NrmGrpcngp 22295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960  ax-pre-sup 9961
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-er 7690  df-map 7807  df-en 7903  df-dom 7904  df-sdom 7905  df-sup 8295  df-inf 8296  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-div 10632  df-nn 10968  df-2 11026  df-n0 11240  df-z 11325  df-uz 11635  df-q 11736  df-rp 11780  df-xneg 11893  df-xadd 11894  df-xmul 11895  df-0g 16026  df-topgen 16028  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-minusg 17350  df-sbg 17351  df-psmet 19660  df-xmet 19661  df-met 19662  df-bl 19663  df-mopn 19664  df-top 20621  df-topon 20638  df-topsp 20651  df-bases 20664  df-xms 22038  df-ms 22039  df-nm 22300  df-ngp 22301
This theorem is referenced by:  sranlm  22401
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