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Theorem nmpropd2 22380
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 2656 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 nmpropd2.5 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
51sqxpeqd 5131 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
65reseq2d 5385 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
74, 6eqtr3d 2656 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
82sqxpeqd 5131 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
98reseq2d 5385 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
107, 9eqtr3d 2656 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11 eqidd 2621 . . . 4 (𝜑𝑎 = 𝑎)
12 nmpropd2.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
131, 2, 12grpidpropd 17242 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
1410, 11, 13oveq123d 6656 . . 3 (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿)))
153, 14mpteq12dv 4724 . 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 2620 . . . 4 (norm‘𝐾) = (norm‘𝐾)
18 eqid 2620 . . . 4 (Base‘𝐾) = (Base‘𝐾)
19 eqid 2620 . . . 4 (0g𝐾) = (0g𝐾)
20 eqid 2620 . . . 4 (dist‘𝐾) = (dist‘𝐾)
21 eqid 2620 . . . 4 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
2217, 18, 19, 20, 21nmfval2 22376 . . 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 17418 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2516, 24mpbid 222 . . 3 (𝜑𝐿 ∈ Grp)
26 eqid 2620 . . . 4 (norm‘𝐿) = (norm‘𝐿)
27 eqid 2620 . . . 4 (Base‘𝐿) = (Base‘𝐿)
28 eqid 2620 . . . 4 (0g𝐿) = (0g𝐿)
29 eqid 2620 . . . 4 (dist‘𝐿) = (dist‘𝐿)
30 eqid 2620 . . . 4 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
3126, 27, 28, 29, 30nmfval2 22376 . . 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 2664 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  cmpt 4720   × cxp 5102  cres 5106  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  distcds 15931  0gc0g 16081  Grpcgrp 17403  normcnm 22362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-grp 17406  df-nm 22368
This theorem is referenced by:  ngppropd  22422
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