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Theorem nmpropd 22617
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 2771 . . . 4 (𝜑𝑥 = 𝑥)
4 eqidd 2771 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5 nmpropd.2 . . . . . 6 (𝜑 → (+g𝐾) = (+g𝐿))
65oveqdr 6818 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
74, 1, 6grpidpropd 17468 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
82, 3, 7oveq123d 6813 . . 3 (𝜑 → (𝑥(dist‘𝐾)(0g𝐾)) = (𝑥(dist‘𝐿)(0g𝐿)))
91, 8mpteq12dv 4865 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿))))
10 eqid 2770 . . 3 (norm‘𝐾) = (norm‘𝐾)
11 eqid 2770 . . 3 (Base‘𝐾) = (Base‘𝐾)
12 eqid 2770 . . 3 (0g𝐾) = (0g𝐾)
13 eqid 2770 . . 3 (dist‘𝐾) = (dist‘𝐾)
1410, 11, 12, 13nmfval 22612 . 2 (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾)))
15 eqid 2770 . . 3 (norm‘𝐿) = (norm‘𝐿)
16 eqid 2770 . . 3 (Base‘𝐿) = (Base‘𝐿)
17 eqid 2770 . . 3 (0g𝐿) = (0g𝐿)
18 eqid 2770 . . 3 (dist‘𝐿) = (dist‘𝐿)
1915, 16, 17, 18nmfval 22612 . 2 (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿)))
209, 14, 193eqtr4g 2829 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  cmpt 4861  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  distcds 16157  0gc0g 16307  normcnm 22600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-0g 16309  df-nm 22606
This theorem is referenced by:  sranlm  22707  rlmnm  22712  zlmnm  30344
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