Theorem isnmhm 24794

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 24758	. . 3 ⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
2	1	elmpocl 7632	. 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3		oveq12 7400	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 LMHom 𝑡) = (𝑆 LMHom 𝑇))
4		oveq12 7400	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 NGHom 𝑡) = (𝑆 NGHom 𝑇))
5	3, 4	ineq12d 4171	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
6		ovex 7424	. . . . . 6 ⊢ (𝑆 LMHom 𝑇) ∈ V
7	6	inex1 5270	. . . . 5 ⊢ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ∈ V
8	5, 1, 7	ovmpoa 7546	. . . 4 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑆 NMHom 𝑇) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
9	8	eleq2d 2847	. . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ 𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇))))
10		elin 3918	. . 3 ⊢ (𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))
11	9, 10	bitrdi 289	. 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
12	2, 11	biadanii 831	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∩ cin 3901  (class class class)co 7391   LMHom clmhm 21074  NrmModcnlm 24628   NGHom cnghm 24754   NMHom cnmhm 24755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-nmhm 24758

This theorem is referenced by:  nmhmrcl1  24795  nmhmrcl2  24796  nmhmlmhm  24797  nmhmnghm  24798  isnmhm2  24800  idnmhm  24802  0nmhm  24803  nmhmco  24804  nmhmplusg  24805  nmhmcn  25170