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Theorem nmpropd 24447
Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
nmpropd.1 (๐œ‘ โ†’ (Baseโ€˜๐พ) = (Baseโ€˜๐ฟ))
nmpropd.2 (๐œ‘ โ†’ (+gโ€˜๐พ) = (+gโ€˜๐ฟ))
nmpropd.3 (๐œ‘ โ†’ (distโ€˜๐พ) = (distโ€˜๐ฟ))
Assertion
Ref Expression
nmpropd (๐œ‘ โ†’ (normโ€˜๐พ) = (normโ€˜๐ฟ))

Proof of Theorem nmpropd
Dummy variables ๐‘ฅ ๐‘ฆ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nmpropd.1 . . 3 (๐œ‘ โ†’ (Baseโ€˜๐พ) = (Baseโ€˜๐ฟ))
2 nmpropd.3 . . . 4 (๐œ‘ โ†’ (distโ€˜๐พ) = (distโ€˜๐ฟ))
3 eqidd 2725 . . . 4 (๐œ‘ โ†’ ๐‘ฅ = ๐‘ฅ)
4 eqidd 2725 . . . . 5 (๐œ‘ โ†’ (Baseโ€˜๐พ) = (Baseโ€˜๐พ))
5 nmpropd.2 . . . . . 6 (๐œ‘ โ†’ (+gโ€˜๐พ) = (+gโ€˜๐ฟ))
65oveqdr 7430 . . . . 5 ((๐œ‘ โˆง (๐‘ฅ โˆˆ (Baseโ€˜๐พ) โˆง ๐‘ฆ โˆˆ (Baseโ€˜๐พ))) โ†’ (๐‘ฅ(+gโ€˜๐พ)๐‘ฆ) = (๐‘ฅ(+gโ€˜๐ฟ)๐‘ฆ))
74, 1, 6grpidpropd 18591 . . . 4 (๐œ‘ โ†’ (0gโ€˜๐พ) = (0gโ€˜๐ฟ))
82, 3, 7oveq123d 7423 . . 3 (๐œ‘ โ†’ (๐‘ฅ(distโ€˜๐พ)(0gโ€˜๐พ)) = (๐‘ฅ(distโ€˜๐ฟ)(0gโ€˜๐ฟ)))
91, 8mpteq12dv 5230 . 2 (๐œ‘ โ†’ (๐‘ฅ โˆˆ (Baseโ€˜๐พ) โ†ฆ (๐‘ฅ(distโ€˜๐พ)(0gโ€˜๐พ))) = (๐‘ฅ โˆˆ (Baseโ€˜๐ฟ) โ†ฆ (๐‘ฅ(distโ€˜๐ฟ)(0gโ€˜๐ฟ))))
10 eqid 2724 . . 3 (normโ€˜๐พ) = (normโ€˜๐พ)
11 eqid 2724 . . 3 (Baseโ€˜๐พ) = (Baseโ€˜๐พ)
12 eqid 2724 . . 3 (0gโ€˜๐พ) = (0gโ€˜๐พ)
13 eqid 2724 . . 3 (distโ€˜๐พ) = (distโ€˜๐พ)
1410, 11, 12, 13nmfval 24441 . 2 (normโ€˜๐พ) = (๐‘ฅ โˆˆ (Baseโ€˜๐พ) โ†ฆ (๐‘ฅ(distโ€˜๐พ)(0gโ€˜๐พ)))
15 eqid 2724 . . 3 (normโ€˜๐ฟ) = (normโ€˜๐ฟ)
16 eqid 2724 . . 3 (Baseโ€˜๐ฟ) = (Baseโ€˜๐ฟ)
17 eqid 2724 . . 3 (0gโ€˜๐ฟ) = (0gโ€˜๐ฟ)
18 eqid 2724 . . 3 (distโ€˜๐ฟ) = (distโ€˜๐ฟ)
1915, 16, 17, 18nmfval 24441 . 2 (normโ€˜๐ฟ) = (๐‘ฅ โˆˆ (Baseโ€˜๐ฟ) โ†ฆ (๐‘ฅ(distโ€˜๐ฟ)(0gโ€˜๐ฟ)))
209, 14, 193eqtr4g 2789 1 (๐œ‘ โ†’ (normโ€˜๐พ) = (normโ€˜๐ฟ))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 395   = wceq 1533   โˆˆ wcel 2098   โ†ฆ cmpt 5222  โ€˜cfv 6534  (class class class)co 7402  Basecbs 17149  +gcplusg 17202  distcds 17211  0gc0g 17390  normcnm 24429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-0g 17392  df-nm 24435
This theorem is referenced by:  sranlm  24545  rlmnm  24550  zlmnm  33465
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