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Mirrors > Home > MPE Home > Th. List > nmhmrcl2 | Structured version Visualization version GIF version |
Description: Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmhmrcl2 | ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnmhm 24782 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 (class class class)co 7430 LMHom clmhm 21035 NrmModcnlm 24608 NGHom cnghm 24742 NMHom cnmhm 24743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-nmhm 24746 |
This theorem is referenced by: nmhmco 24792 nmhmplusg 24793 |
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