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Theorem nmhmrcl1 24109
Description: Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
Assertion
Ref Expression
nmhmrcl1 (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)

Proof of Theorem nmhmrcl1
StepHypRef Expression
1 isnmhm 24108 . . 3 (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
21simplbi 498 . 2 (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
32simpld 495 1 (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  (class class class)co 7356   LMHom clmhm 20478  NrmModcnlm 23934   NGHom cnghm 24068   NMHom cnmhm 24069
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-nmhm 24072
This theorem is referenced by:  nmhmco  24118  nmhmplusg  24119
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