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Theorem isnmhm 23358
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.
StepHypRef Expression
1 df-nmhm 23322 . . 3 NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
21elmpocl 7381 . 2 (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3 oveq12 7158 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 LMHom 𝑡) = (𝑆 LMHom 𝑇))
4 oveq12 7158 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 NGHom 𝑡) = (𝑆 NGHom 𝑇))
53, 4ineq12d 4175 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
6 ovex 7182 . . . . . 6 (𝑆 LMHom 𝑇) ∈ V
76inex1 5207 . . . . 5 ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ∈ V
85, 1, 7ovmpoa 7298 . . . 4 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑆 NMHom 𝑇) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
98eleq2d 2901 . . 3 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ 𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇))))
10 elin 3935 . . 3 (𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))
119, 10syl6bb 290 . 2 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
122, 11biadanii 821 1 (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3918  (class class class)co 7149   LMHom clmhm 19791  NrmModcnlm 23193   NGHom cnghm 23318   NMHom cnmhm 23319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-nmhm 23322
This theorem is referenced by:  nmhmrcl1  23359  nmhmrcl2  23360  nmhmlmhm  23361  nmhmnghm  23362  isnmhm2  23364  idnmhm  23366  0nmhm  23367  nmhmco  23368  nmhmplusg  23369  nmhmcn  23731
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