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Theorem nmpropd 24623
Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
nmpropd.1 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
nmpropd.2 (𝜑 → (+g𝐾) = (+g𝐿))
nmpropd.3 (𝜑 → (dist‘𝐾) = (dist‘𝐿))
Assertion
Ref Expression
nmpropd (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Proof of Theorem nmpropd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmpropd.1 . . 3 (𝜑 → (Base‘𝐾) = (Base‘𝐿))
2 nmpropd.3 . . . 4 (𝜑 → (dist‘𝐾) = (dist‘𝐿))
3 eqidd 2736 . . . 4 (𝜑𝑥 = 𝑥)
4 eqidd 2736 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5 nmpropd.2 . . . . . 6 (𝜑 → (+g𝐾) = (+g𝐿))
65oveqdr 7459 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
74, 1, 6grpidpropd 18688 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
82, 3, 7oveq123d 7452 . . 3 (𝜑 → (𝑥(dist‘𝐾)(0g𝐾)) = (𝑥(dist‘𝐿)(0g𝐿)))
91, 8mpteq12dv 5239 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿))))
10 eqid 2735 . . 3 (norm‘𝐾) = (norm‘𝐾)
11 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
12 eqid 2735 . . 3 (0g𝐾) = (0g𝐾)
13 eqid 2735 . . 3 (dist‘𝐾) = (dist‘𝐾)
1410, 11, 12, 13nmfval 24617 . 2 (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾)))
15 eqid 2735 . . 3 (norm‘𝐿) = (norm‘𝐿)
16 eqid 2735 . . 3 (Base‘𝐿) = (Base‘𝐿)
17 eqid 2735 . . 3 (0g𝐿) = (0g𝐿)
18 eqid 2735 . . 3 (dist‘𝐿) = (dist‘𝐿)
1915, 16, 17, 18nmfval 24617 . 2 (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿)))
209, 14, 193eqtr4g 2800 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  distcds 17307  0gc0g 17486  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-rab 3434  df-v 3480  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-ov 7434  df-0g 17488  df-nm 24611
This theorem is referenced by:  sranlm  24721  rlmnm  24726  zlmnm  33927
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