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Theorem nmpropd 24628
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 2741 . . . 4 (𝜑𝑥 = 𝑥)
4 eqidd 2741 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5 nmpropd.2 . . . . . 6 (𝜑 → (+g𝐾) = (+g𝐿))
65oveqdr 7476 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
74, 1, 6grpidpropd 18700 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
82, 3, 7oveq123d 7469 . . 3 (𝜑 → (𝑥(dist‘𝐾)(0g𝐾)) = (𝑥(dist‘𝐿)(0g𝐿)))
91, 8mpteq12dv 5257 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿))))
10 eqid 2740 . . 3 (norm‘𝐾) = (norm‘𝐾)
11 eqid 2740 . . 3 (Base‘𝐾) = (Base‘𝐾)
12 eqid 2740 . . 3 (0g𝐾) = (0g𝐾)
13 eqid 2740 . . 3 (dist‘𝐾) = (dist‘𝐾)
1410, 11, 12, 13nmfval 24622 . 2 (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾)))
15 eqid 2740 . . 3 (norm‘𝐿) = (norm‘𝐿)
16 eqid 2740 . . 3 (Base‘𝐿) = (Base‘𝐿)
17 eqid 2740 . . 3 (0g𝐿) = (0g𝐿)
18 eqid 2740 . . 3 (dist‘𝐿) = (dist‘𝐿)
1915, 16, 17, 18nmfval 24622 . 2 (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿)))
209, 14, 193eqtr4g 2805 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  cmpt 5249  cfv 6573  (class class class)co 7448  Basecbs 17258  +gcplusg 17311  distcds 17320  0gc0g 17499  normcnm 24610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-0g 17501  df-nm 24616
This theorem is referenced by:  sranlm  24726  rlmnm  24731  zlmnm  33912
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