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Theorem nmpropd2 23199
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 2859 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4 nmpropd2.5 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))
51sqxpeqd 5564 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐾) × (Base‘𝐾)))
65reseq2d 5831 . . . . . 6 (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
74, 6eqtr3d 2859 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))))
82sqxpeqd 5564 . . . . . 6 (𝜑 → (𝐵 × 𝐵) = ((Base‘𝐿) × (Base‘𝐿)))
98reseq2d 5831 . . . . 5 (𝜑 → ((dist‘𝐿) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
107, 9eqtr3d 2859 . . . 4 (𝜑 → ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))))
11 eqidd 2823 . . . 4 (𝜑𝑎 = 𝑎)
12 nmpropd2.4 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
131, 2, 12grpidpropd 17863 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
1410, 11, 13oveq123d 7161 . . 3 (𝜑 → (𝑎((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))(0g𝐾)) = (𝑎((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))(0g𝐿)))
153, 14mpteq12dv 5127 . 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 2822 . . . 4 (norm‘𝐾) = (norm‘𝐾)
18 eqid 2822 . . . 4 (Base‘𝐾) = (Base‘𝐾)
19 eqid 2822 . . . 4 (0g𝐾) = (0g𝐾)
20 eqid 2822 . . . 4 (dist‘𝐾) = (dist‘𝐾)
21 eqid 2822 . . . 4 ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾))) = ((dist‘𝐾) ↾ ((Base‘𝐾) × (Base‘𝐾)))
2217, 18, 19, 20, 21nmfval2 23195 . . 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 18109 . . . 4 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
2516, 24mpbid 235 . . 3 (𝜑𝐿 ∈ Grp)
26 eqid 2822 . . . 4 (norm‘𝐿) = (norm‘𝐿)
27 eqid 2822 . . . 4 (Base‘𝐿) = (Base‘𝐿)
28 eqid 2822 . . . 4 (0g𝐿) = (0g𝐿)
29 eqid 2822 . . . 4 (dist‘𝐿) = (dist‘𝐿)
30 eqid 2822 . . . 4 ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿))) = ((dist‘𝐿) ↾ ((Base‘𝐿) × (Base‘𝐿)))
3126, 27, 28, 29, 30nmfval2 23195 . . 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 2867 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cmpt 5122   × cxp 5530  cres 5534  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  distcds 16565  0gc0g 16704  Grpcgrp 18094  normcnm 23181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-riota 7098  df-ov 7143  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-grp 18097  df-nm 23187
This theorem is referenced by:  ngppropd  23241
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