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Mirrors > Home > MPE Home > Th. List > isnghm | Structured version Visualization version GIF version |
Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.) |
Ref | Expression |
---|---|
nmofval.1 | ⊢ 𝑁 = (𝑆 normOp 𝑇) |
Ref | Expression |
---|---|
isnghm | ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmofval.1 | . . . 4 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | 1 | nghmfval 24758 | . . 3 ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) |
3 | 2 | eleq2i 2830 | . 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ 𝐹 ∈ (◡𝑁 “ ℝ)) |
4 | n0i 4345 | . . . 4 ⊢ (𝐹 ∈ (◡𝑁 “ ℝ) → ¬ (◡𝑁 “ ℝ) = ∅) | |
5 | nmoffn 24747 | . . . . . . . . . . 11 ⊢ normOp Fn (NrmGrp × NrmGrp) | |
6 | 5 | fndmi 6672 | . . . . . . . . . 10 ⊢ dom normOp = (NrmGrp × NrmGrp) |
7 | 6 | ndmov 7616 | . . . . . . . . 9 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅) |
8 | 1, 7 | eqtrid 2786 | . . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅) |
9 | 8 | cnveqd 5888 | . . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅) |
10 | cnv0 6162 | . . . . . . 7 ⊢ ◡∅ = ∅ | |
11 | 9, 10 | eqtrdi 2790 | . . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅) |
12 | 11 | imaeq1d 6078 | . . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ)) |
13 | 0ima 6097 | . . . . 5 ⊢ (∅ “ ℝ) = ∅ | |
14 | 12, 13 | eqtrdi 2790 | . . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅) |
15 | 4, 14 | nsyl2 141 | . . 3 ⊢ (𝐹 ∈ (◡𝑁 “ ℝ) → (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp)) |
16 | 1 | nmof 24755 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁:(𝑆 GrpHom 𝑇)⟶ℝ*) |
17 | ffn 6736 | . . . 4 ⊢ (𝑁:(𝑆 GrpHom 𝑇)⟶ℝ* → 𝑁 Fn (𝑆 GrpHom 𝑇)) | |
18 | elpreima 7077 | . . . 4 ⊢ (𝑁 Fn (𝑆 GrpHom 𝑇) → (𝐹 ∈ (◡𝑁 “ ℝ) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ))) | |
19 | 16, 17, 18 | 3syl 18 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝐹 ∈ (◡𝑁 “ ℝ) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ))) |
20 | 15, 19 | biadanii 822 | . 2 ⊢ (𝐹 ∈ (◡𝑁 “ ℝ) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ))) |
21 | 3, 20 | bitri 275 | 1 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∅c0 4338 × cxp 5686 ◡ccnv 5687 “ cima 5691 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 ℝ*cxr 11291 GrpHom cghm 19242 NrmGrpcngp 24605 normOp cnmo 24741 NGHom cnghm 24742 |
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-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 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-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-ico 13389 df-nmo 24744 df-nghm 24745 |
This theorem is referenced by: isnghm2 24760 nghmcl 24763 nmoi 24764 nghmrcl1 24768 nghmrcl2 24769 nghmghm 24770 isnmhm2 24788 |
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