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Mirrors > Home > MPE Home > Th. List > isnmhm | Structured version Visualization version GIF version |
Description: A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
isnmhm | ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nmhm 23246 | . . 3 ⊢ NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡))) | |
2 | 1 | elmpocl 7376 | . 2 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod)) |
3 | oveq12 7154 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 LMHom 𝑡) = (𝑆 LMHom 𝑇)) | |
4 | oveq12 7154 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 NGHom 𝑡) = (𝑆 NGHom 𝑇)) | |
5 | 3, 4 | ineq12d 4187 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇))) |
6 | ovex 7178 | . . . . . 6 ⊢ (𝑆 LMHom 𝑇) ∈ V | |
7 | 6 | inex1 5212 | . . . . 5 ⊢ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ∈ V |
8 | 5, 1, 7 | ovmpoa 7294 | . . . 4 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝑆 NMHom 𝑇) = ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇))) |
9 | 8 | eleq2d 2895 | . . 3 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ 𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)))) |
10 | elin 4166 | . . 3 ⊢ (𝐹 ∈ ((𝑆 LMHom 𝑇) ∩ (𝑆 NGHom 𝑇)) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))) | |
11 | 9, 10 | syl6bb 288 | . 2 ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))) |
12 | 2, 11 | biadanii 818 | 1 ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇)))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 (class class class)co 7145 LMHom clmhm 19720 NrmModcnlm 23117 NGHom cnghm 23242 NMHom cnmhm 23243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-nmhm 23246 |
This theorem is referenced by: nmhmrcl1 23283 nmhmrcl2 23284 nmhmlmhm 23285 nmhmnghm 23286 isnmhm2 23288 idnmhm 23290 0nmhm 23291 nmhmco 23292 nmhmplusg 23293 nmhmcn 23651 |
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