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Theorem nmpropd2 24608
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 2779 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 nmpropd2.5 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
51sqxpeqd 5717 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
65reseq2d 5997 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
74, 6eqtr3d 2779 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
82sqxpeqd 5717 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
98reseq2d 5997 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
107, 9eqtr3d 2779 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11 eqidd 2738 . . . 4 (𝜑𝑎 = 𝑎)
12 nmpropd2.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
131, 2, 12grpidpropd 18675 . . . 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 5233 . 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 2737 . . . 4 (norm‘𝐾) = (norm‘𝐾)
18 eqid 2737 . . . 4 (Base‘𝐾) = (Base‘𝐾)
19 eqid 2737 . . . 4 (0g𝐾) = (0g𝐾)
20 eqid 2737 . . . 4 (dist‘𝐾) = (dist‘𝐾)
21 eqid 2737 . . . 4 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
2217, 18, 19, 20, 21nmfval2 24604 . . 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 18969 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2516, 24mpbid 232 . . 3 (𝜑𝐿 ∈ Grp)
26 eqid 2737 . . . 4 (norm‘𝐿) = (norm‘𝐿)
27 eqid 2737 . . . 4 (Base‘𝐿) = (Base‘𝐿)
28 eqid 2737 . . . 4 (0g𝐿) = (0g𝐿)
29 eqid 2737 . . . 4 (dist‘𝐿) = (dist‘𝐿)
30 eqid 2737 . . . 4 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
3126, 27, 28, 29, 30nmfval2 24604 . . 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 2787 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cmpt 5225   × cxp 5683  cres 5687  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  distcds 17306  0gc0g 17484  Grpcgrp 18951  normcnm 24589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-nm 24595
This theorem is referenced by:  ngppropd  24650
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