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Theorem nmpropd 22392
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 2622 . . . 4 (𝜑𝑥 = 𝑥)
4 eqidd 2622 . . . . 5 (𝜑 → (Base‘𝐾) = (Base‘𝐾))
5 nmpropd.2 . . . . . 6 (𝜑 → (+g𝐾) = (+g𝐿))
65oveqdr 6671 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
74, 1, 6grpidpropd 17255 . . . 4 (𝜑 → (0g𝐾) = (0g𝐿))
82, 3, 7oveq123d 6668 . . 3 (𝜑 → (𝑥(dist‘𝐾)(0g𝐾)) = (𝑥(dist‘𝐿)(0g𝐿)))
91, 8mpteq12dv 4731 . 2 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾))) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿))))
10 eqid 2621 . . 3 (norm‘𝐾) = (norm‘𝐾)
11 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
12 eqid 2621 . . 3 (0g𝐾) = (0g𝐾)
13 eqid 2621 . . 3 (dist‘𝐾) = (dist‘𝐾)
1410, 11, 12, 13nmfval 22387 . 2 (norm‘𝐾) = (𝑥 ∈ (Base‘𝐾) ↦ (𝑥(dist‘𝐾)(0g𝐾)))
15 eqid 2621 . . 3 (norm‘𝐿) = (norm‘𝐿)
16 eqid 2621 . . 3 (Base‘𝐿) = (Base‘𝐿)
17 eqid 2621 . . 3 (0g𝐿) = (0g𝐿)
18 eqid 2621 . . 3 (dist‘𝐿) = (dist‘𝐿)
1915, 16, 17, 18nmfval 22387 . 2 (norm‘𝐿) = (𝑥 ∈ (Base‘𝐿) ↦ (𝑥(dist‘𝐿)(0g𝐿)))
209, 14, 193eqtr4g 2680 1 (𝜑 → (norm‘𝐾) = (norm‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989  cmpt 4727  cfv 5886  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  distcds 15944  0gc0g 16094  normcnm 22375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-0g 16096  df-nm 22381
This theorem is referenced by:  sranlm  22482  rlmnm  22487  zlmnm  29995
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