Theorem nmhmrcl2 24620

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

Step	Hyp	Ref	Expression
1		isnmhm 24618	. . 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	simprd 495	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2098  (class class class)co 7405   LMHom clmhm 20867  NrmModcnlm 24444   NGHom cnghm 24578   NMHom cnmhm 24579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-nmhm 24582

This theorem is referenced by:  nmhmco  24628  nmhmplusg  24629