Theorem nmpropd 22398

Description: Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)

Hypotheses

Ref	Expression
nmpropd.1	|- ( ph -> ( Base ` K ) = ( Base ` L ) )
nmpropd.2	|- ( ph -> ( +g ` K ) = ( +g ` L ) )
nmpropd.3	|- ( ph -> ( dist ` K ) = ( dist ` L ) )

Assertion

Ref	Expression
nmpropd	|- ( ph -> ( norm ` K ) = ( norm ` L ) )

Proof of Theorem nmpropd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nmpropd.1	. . 3 |- ( ph -> ( Base ` K ) = ( Base ` L ) )
2		nmpropd.3	. . . 4 |- ( ph -> ( dist ` K ) = ( dist ` L ) )
3		eqidd 2623	. . . 4 |- ( ph -> x = x )
4		eqidd 2623	. . . . 5 |- ( ph -> ( Base ` K ) = ( Base ` K ) )
5		nmpropd.2	. . . . . 6 |- ( ph -> ( +g ` K ) = ( +g ` L ) )
6	5	oveqdr 6674	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` K ) /\ y e. ( Base ` K ) ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )
7	4, 1, 6	grpidpropd 17261	. . . 4 |- ( ph -> ( 0g ` K ) = ( 0g ` L ) )
8	2, 3, 7	oveq123d 6671	. . 3 |- ( ph -> ( x ( dist ` K ) ( 0g ` K ) ) = ( x ( dist ` L ) ( 0g ` L ) ) )
9	1, 8	mpteq12dv 4733	. 2 |- ( ph -> ( x e. ( Base ` K ) |-> ( x ( dist ` K ) ( 0g ` K ) ) ) = ( x e. ( Base ` L ) |-> ( x ( dist ` L ) ( 0g ` L ) ) ) )
10		eqid 2622	. . 3 |- ( norm ` K ) = ( norm ` K )
11		eqid 2622	. . 3 |- ( Base ` K ) = ( Base ` K )
12		eqid 2622	. . 3 |- ( 0g ` K ) = ( 0g ` K )
13		eqid 2622	. . 3 |- ( dist ` K ) = ( dist ` K )
14	10, 11, 12, 13	nmfval 22393	. 2 |- ( norm ` K ) = ( x e. ( Base ` K ) |-> ( x ( dist ` K ) ( 0g ` K ) ) )
15		eqid 2622	. . 3 |- ( norm ` L ) = ( norm ` L )
16		eqid 2622	. . 3 |- ( Base ` L ) = ( Base ` L )
17		eqid 2622	. . 3 |- ( 0g ` L ) = ( 0g ` L )
18		eqid 2622	. . 3 |- ( dist ` L ) = ( dist ` L )
19	15, 16, 17, 18	nmfval 22393	. 2 |- ( norm ` L ) = ( x e. ( Base ` L ) |-> ( x ( dist ` L ) ( 0g ` L ) ) )
20	9, 14, 19	3eqtr4g 2681	1 |- ( ph -> ( norm ` K ) = ( norm ` L ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   distcds 15950   0gc0g 16100   normcnm 22381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-0g 16102  df-nm 22387

This theorem is referenced by:  sranlm  22488  rlmnm  22493  zlmnm  30010