MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnmhm Structured version   Visualization version   GIF version

Theorem isnmhm 24782
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 24746 . . 3 NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))
21elmpocl 7673 . 2 (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3 oveq12 7439 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 LMHom 𝑡) = (𝑆 LMHom 𝑇))
4 oveq12 7439 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 NGHom 𝑡) = (𝑆 NGHom 𝑇))
53, 4ineq12d 4228 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
6 ovex 7463 . . . . . 6 (𝑆 LMHom 𝑇) ∈ V
76inex1 5322 . . . . 5 ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ∈ V
85, 1, 7ovmpoa 7587 . . . 4 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑆 NMHom 𝑇) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))
98eleq2d 2824 . . 3 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ 𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇))))
10 elin 3978 . . 3 (𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))
119, 10bitrdi 287 . 2 ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
122, 11biadanii 822 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 206  wa 395   = wceq 1536  wcel 2105  cin 3961  (class class class)co 7430   LMHom clmhm 21035  NrmModcnlm 24608   NGHom cnghm 24742   NMHom cnmhm 24743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-nmhm 24746
This theorem is referenced by:  nmhmrcl1  24783  nmhmrcl2  24784  nmhmlmhm  24785  nmhmnghm  24786  isnmhm2  24788  idnmhm  24790  0nmhm  24791  nmhmco  24792  nmhmplusg  24793  nmhmcn  25166
  Copyright terms: Public domain W3C validator