Theorem isnmhm 24610

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 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (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 24574	. . 3 ⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
2	1	elmpocl 7610	. 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3		oveq12 7378	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 LMHom 𝑡) = (𝑆 LMHom 𝑇))
4		oveq12 7378	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 NGHom 𝑡) = (𝑆 NGHom 𝑇))
5	3, 4	ineq12d 4180	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
6		ovex 7402	. . . . . 6 ⊢ (𝑆 LMHom 𝑇) ∈ V
7	6	inex1 5267	. . . . 5 ⊢ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ∈ V
8	5, 1, 7	ovmpoa 7524	. . . 4 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑆 NMHom 𝑇) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
9	8	eleq2d 2814	. . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ 𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇))))
10		elin 3927	. . 3 ⊢ (𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))
11	9, 10	bitrdi 287	. 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
12	2, 11	biadanii 821	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3910  (class class class)co 7369   LMHom clmhm 20902  NrmModcnlm 24444   NGHom cnghm 24570   NMHom cnmhm 24571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-nmhm 24574

This theorem is referenced by:  nmhmrcl1  24611  nmhmrcl2  24612  nmhmlmhm  24613  nmhmnghm  24614  isnmhm2  24616  idnmhm  24618  0nmhm  24619  nmhmco  24620  nmhmplusg  24621  nmhmcn  24996