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Theorem nmpropd2 24624
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 2777 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 nmpropd2.5 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
51sqxpeqd 5721 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
65reseq2d 6000 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
74, 6eqtr3d 2777 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
82sqxpeqd 5721 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
98reseq2d 6000 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
107, 9eqtr3d 2777 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11 eqidd 2736 . . . 4 (𝜑𝑎 = 𝑎)
12 nmpropd2.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
131, 2, 12grpidpropd 18688 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
1410, 11, 13oveq123d 7452 . . 3 (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿)))
153, 14mpteq12dv 5239 . 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 2735 . . . 4 (norm‘𝐾) = (norm‘𝐾)
18 eqid 2735 . . . 4 (Base‘𝐾) = (Base‘𝐾)
19 eqid 2735 . . . 4 (0g𝐾) = (0g𝐾)
20 eqid 2735 . . . 4 (dist‘𝐾) = (dist‘𝐾)
21 eqid 2735 . . . 4 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
2217, 18, 19, 20, 21nmfval2 24620 . . 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 18982 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2516, 24mpbid 232 . . 3 (𝜑𝐿 ∈ Grp)
26 eqid 2735 . . . 4 (norm‘𝐿) = (norm‘𝐿)
27 eqid 2735 . . . 4 (Base‘𝐿) = (Base‘𝐿)
28 eqid 2735 . . . 4 (0g𝐿) = (0g𝐿)
29 eqid 2735 . . . 4 (dist‘𝐿) = (dist‘𝐿)
30 eqid 2735 . . . 4 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
3126, 27, 28, 29, 30nmfval2 24620 . . 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 2785 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231   × cxp 5687  cres 5691  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  distcds 17307  0gc0g 17486  Grpcgrp 18964  normcnm 24605
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-nm 24611
This theorem is referenced by:  ngppropd  24666
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