Theorem isnmhm 22550

Description: A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)

Assertion

Ref	Expression
isnmhm	|- ( F e. ( S NMHom T ) <-> ( ( S e. NrmMod /\ T e. NrmMod ) /\ ( F e. ( S LMHom T ) /\ F e. ( S NGHom T ) ) ) )

Proof of Theorem isnmhm

Dummy variables t s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nmhm 22514	. . 3 |- NMHom = ( s e. NrmMod , t e. NrmMod |-> ( ( s LMHom t ) i^i ( s NGHom t ) ) )
2	1	elmpt2cl 6876	. 2 |- ( F e. ( S NMHom T ) -> ( S e. NrmMod /\ T e. NrmMod ) )
3		oveq12 6659	. . . . . 6 |- ( ( s = S /\ t = T ) -> ( s LMHom t ) = ( S LMHom T ) )
4		oveq12 6659	. . . . . 6 |- ( ( s = S /\ t = T ) -> ( s NGHom t ) = ( S NGHom T ) )
5	3, 4	ineq12d 3815	. . . . 5 |- ( ( s = S /\ t = T ) -> ( ( s LMHom t ) i^i ( s NGHom t ) ) = ( ( S LMHom T ) i^i ( S NGHom T ) ) )
6		ovex 6678	. . . . . 6 |- ( S LMHom T ) e. _V
7	6	inex1 4799	. . . . 5 |- ( ( S LMHom T ) i^i ( S NGHom T ) ) e. _V
8	5, 1, 7	ovmpt2a 6791	. . . 4 |- ( ( S e. NrmMod /\ T e. NrmMod ) -> ( S NMHom T ) = ( ( S LMHom T ) i^i ( S NGHom T ) ) )
9	8	eleq2d 2687	. . 3 |- ( ( S e. NrmMod /\ T e. NrmMod ) -> ( F e. ( S NMHom T ) <-> F e. ( ( S LMHom T ) i^i ( S NGHom T ) ) ) )
10		elin 3796	. . 3 |- ( F e. ( ( S LMHom T ) i^i ( S NGHom T ) ) <-> ( F e. ( S LMHom T ) /\ F e. ( S NGHom T ) ) )
11	9, 10	syl6bb 276	. 2 |- ( ( S e. NrmMod /\ T e. NrmMod ) -> ( F e. ( S NMHom T ) <-> ( F e. ( S LMHom T ) /\ F e. ( S NGHom T ) ) ) )
12	2, 11	biadan2 674	1 |- ( F e. ( S NMHom T ) <-> ( ( S e. NrmMod /\ T e. NrmMod ) /\ ( F e. ( S LMHom T ) /\ F e. ( S NGHom T ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573  (class class class)co 6650   LMHom clmhm 19019  NrmModcnlm 22385   NGHom cnghm 22510   NMHom cnmhm 22511

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

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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-nmhm 22514

This theorem is referenced by:  nmhmrcl1  22551  nmhmrcl2  22552  nmhmlmhm  22553  nmhmnghm  22554  isnmhm2  22556  idnmhm  22558  0nmhm  22559  nmhmco  22560  nmhmplusg  22561  nmhmcn  22920