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Theorem nmpropd 23200
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 2799 . . . 4 (𝜑𝑥 = 𝑥)
4 eqidd 2799 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5 nmpropd.2 . . . . . 6 (𝜑 → (+g𝐾) = (+g𝐿))
65oveqdr 7163 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
74, 1, 6grpidpropd 17864 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
82, 3, 7oveq123d 7156 . . 3 (𝜑 → (𝑥(dist‘𝐾)(0g𝐾)) = (𝑥(dist‘𝐿)(0g𝐿)))
91, 8mpteq12dv 5115 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿))))
10 eqid 2798 . . 3 (norm‘𝐾) = (norm‘𝐾)
11 eqid 2798 . . 3 (Base‘𝐾) = (Base‘𝐾)
12 eqid 2798 . . 3 (0g𝐾) = (0g𝐾)
13 eqid 2798 . . 3 (dist‘𝐾) = (dist‘𝐾)
1410, 11, 12, 13nmfval 23195 . 2 (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾)))
15 eqid 2798 . . 3 (norm‘𝐿) = (norm‘𝐿)
16 eqid 2798 . . 3 (Base‘𝐿) = (Base‘𝐿)
17 eqid 2798 . . 3 (0g𝐿) = (0g𝐿)
18 eqid 2798 . . 3 (dist‘𝐿) = (dist‘𝐿)
1915, 16, 17, 18nmfval 23195 . 2 (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿)))
209, 14, 193eqtr4g 2858 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  cmpt 5110  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  distcds 16566  0gc0g 16705  normcnm 23183
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-0g 16707  df-nm 23189
This theorem is referenced by:  sranlm  23290  rlmnm  23295  zlmnm  31317
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