Theorem nmhmlmhm 24917

Description: A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)

Assertion

Ref	Expression
nmhmlmhm	⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))

Proof of Theorem nmhmlmhm

Step	Hyp	Ref	Expression
1		isnmhm 24914	. . 3 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
2	1	simprbi 502	. 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))
3	2	simpld 499	1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  (class class class)co 7412   LMHom clmhm 21151  NrmModcnlm 24748   NGHom cnghm 24874   NMHom cnmhm 24875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-nmhm 24878

This theorem is used by:  nmhmco  24924  nmhmplusg  24925