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Theorem nmhmrcl2 23357
Description: Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
nmhmrcl2 (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)

Proof of Theorem nmhmrcl2
StepHypRef Expression
1 isnmhm 23355 . . 3 (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
21simplbi 501 . 2 (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
32simprd 499 1 (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  (class class class)co 7149   LMHom clmhm 19791  NrmModcnlm 23190   NGHom cnghm 23315   NMHom cnmhm 23316
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 23319
This theorem is referenced by:  nmhmco  23365  nmhmplusg  23366
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