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Theorem nmpropd 23973
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 2734 . . . 4 (๐œ‘ โ†’ ๐‘ฅ = ๐‘ฅ)
4 eqidd 2734 . . . . 5 (๐œ‘ โ†’ (Baseโ€˜๐พ) = (Baseโ€˜๐พ))
5 nmpropd.2 . . . . . 6 (๐œ‘ โ†’ (+gโ€˜๐พ) = (+gโ€˜๐ฟ))
65oveqdr 7389 . . . . 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 18525 . . . 4 (๐œ‘ โ†’ (0gโ€˜๐พ) = (0gโ€˜๐ฟ))
82, 3, 7oveq123d 7382 . . 3 (๐œ‘ โ†’ (๐‘ฅ(distโ€˜๐พ)(0gโ€˜๐พ)) = (๐‘ฅ(distโ€˜๐ฟ)(0gโ€˜๐ฟ)))
91, 8mpteq12dv 5200 . 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 2733 . . 3 (normโ€˜๐พ) = (normโ€˜๐พ)
11 eqid 2733 . . 3 (Baseโ€˜๐พ) = (Baseโ€˜๐พ)
12 eqid 2733 . . 3 (0gโ€˜๐พ) = (0gโ€˜๐พ)
13 eqid 2733 . . 3 (distโ€˜๐พ) = (distโ€˜๐พ)
1410, 11, 12, 13nmfval 23967 . 2 (normโ€˜๐พ) = (๐‘ฅ โˆˆ (Baseโ€˜๐พ) โ†ฆ (๐‘ฅ(distโ€˜๐พ)(0gโ€˜๐พ)))
15 eqid 2733 . . 3 (normโ€˜๐ฟ) = (normโ€˜๐ฟ)
16 eqid 2733 . . 3 (Baseโ€˜๐ฟ) = (Baseโ€˜๐ฟ)
17 eqid 2733 . . 3 (0gโ€˜๐ฟ) = (0gโ€˜๐ฟ)
18 eqid 2733 . . 3 (distโ€˜๐ฟ) = (distโ€˜๐ฟ)
1915, 16, 17, 18nmfval 23967 . 2 (normโ€˜๐ฟ) = (๐‘ฅ โˆˆ (Baseโ€˜๐ฟ) โ†ฆ (๐‘ฅ(distโ€˜๐ฟ)(0gโ€˜๐ฟ)))
209, 14, 193eqtr4g 2798 1 (๐œ‘ โ†’ (normโ€˜๐พ) = (normโ€˜๐ฟ))
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 397   = wceq 1542   โˆˆ wcel 2107   โ†ฆ cmpt 5192  โ€˜cfv 6500  (class class class)co 7361  Basecbs 17091  +gcplusg 17141  distcds 17150  0gc0g 17329  normcnm 23955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-0g 17331  df-nm 23961
This theorem is referenced by:  sranlm  24071  rlmnm  24076  zlmnm  32611
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