Theorem nmhmrcl1 24684

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

Step	Hyp	Ref	Expression
1		isnmhm 24683	. . 3 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
2	1	simplbi 497	. 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3	2	simpld 494	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  (class class class)co 7403   LMHom clmhm 20975  NrmModcnlm 24517   NGHom cnghm 24643   NMHom cnmhm 24644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6483  df-fun 6532  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-nmhm 24647

This theorem is referenced by:  nmhmco  24693  nmhmplusg  24694