Theorem nmhmrcl1 24885

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 24884	. . 3 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
2	1	simplbi 501	. 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod))
3	2	simpld 499	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143  (class class class)co 7412   LMHom clmhm 21121  NrmModcnlm 24718   NGHom cnghm 24844   NMHom cnmhm 24845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-nmhm 24848

This theorem is referenced by:  nmhmco  24894  nmhmplusg  24895