Theorem nmhmrcl2 25097

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 25095	. . 3 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
2	1	simplbi 502	. 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3	2	simprd 501	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147  (class class class)co 7443   LMHom clmhm 21324  NrmModcnlm 24929   NGHom cnghm 25055   NMHom cnmhm 25056

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-nmhm 25059

This theorem is used by:  nmhmco  25105  nmhmplusg  25106