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Theorem nmpropd2 22209
 Description: Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
nmpropd2.1 (𝜑𝐵 = (Base‘𝐾))
nmpropd2.2 (𝜑𝐵 = (Base‘𝐿))
nmpropd2.3 (𝜑𝐾 ∈ Grp)
nmpropd2.4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
nmpropd2.5 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
nmpropd2 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem nmpropd2
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 nmpropd2.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 nmpropd2.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
31, 2eqtr3d 2646 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 nmpropd2.5 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
51sqxpeqd 5065 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
65reseq2d 5317 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
74, 6eqtr3d 2646 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
82sqxpeqd 5065 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
98reseq2d 5317 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
107, 9eqtr3d 2646 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11 eqidd 2611 . . . 4 (𝜑𝑎 = 𝑎)
12 nmpropd2.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
131, 2, 12grpidpropd 17084 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
1410, 11, 13oveq123d 6570 . . 3 (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿)))
153, 14mpteq12dv 4663 . 2 (𝜑 → (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾))) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿))))
16 nmpropd2.3 . . 3 (𝜑𝐾 ∈ Grp)
17 eqid 2610 . . . 4 (norm‘𝐾) = (norm‘𝐾)
18 eqid 2610 . . . 4 (Base‘𝐾) = (Base‘𝐾)
19 eqid 2610 . . . 4 (0g𝐾) = (0g𝐾)
20 eqid 2610 . . . 4 (dist‘𝐾) = (dist‘𝐾)
21 eqid 2610 . . . 4 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
2217, 18, 19, 20, 21nmfval2 22205 . . 3 (𝐾 ∈ Grp → (norm‘𝐾) = (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾))))
2316, 22syl 17 . 2 (𝜑 → (norm‘𝐾) = (𝑎 ∈ (Base‘𝐾) ↦ (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾))))
241, 2, 12grppropd 17260 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2516, 24mpbid 221 . . 3 (𝜑𝐿 ∈ Grp)
26 eqid 2610 . . . 4 (norm‘𝐿) = (norm‘𝐿)
27 eqid 2610 . . . 4 (Base‘𝐿) = (Base‘𝐿)
28 eqid 2610 . . . 4 (0g𝐿) = (0g𝐿)
29 eqid 2610 . . . 4 (dist‘𝐿) = (dist‘𝐿)
30 eqid 2610 . . . 4 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
3126, 27, 28, 29, 30nmfval2 22205 . . 3 (𝐿 ∈ Grp → (norm‘𝐿) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿))))
3225, 31syl 17 . 2 (𝜑 → (norm‘𝐿) = (𝑎 ∈ (Base‘𝐿) ↦ (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿))))
3315, 23, 323eqtr4d 2654 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  distcds 15777  0gc0g 15923  Grpcgrp 17245  normcnm 22191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-nm 22197 This theorem is referenced by:  ngppropd  22251
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