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Mirrors > Home > MPE Home > Th. List > isnghm3 | 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 |
---|---|
isnghm3 | ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) < +∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmofval.1 | . . 3 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
2 | 1 | isnghm2 24661 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ)) |
3 | 1 | nmocl 24657 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ*) |
4 | 1 | nmoge0 24658 | . . . 4 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹)) |
5 | ge0gtmnf 13191 | . . . 4 ⊢ (((𝑁‘𝐹) ∈ ℝ* ∧ 0 ≤ (𝑁‘𝐹)) → -∞ < (𝑁‘𝐹)) | |
6 | 3, 4, 5 | syl2anc 582 | . . 3 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → -∞ < (𝑁‘𝐹)) |
7 | xrrebnd 13187 | . . . 4 ⊢ ((𝑁‘𝐹) ∈ ℝ* → ((𝑁‘𝐹) ∈ ℝ ↔ (-∞ < (𝑁‘𝐹) ∧ (𝑁‘𝐹) < +∞))) | |
8 | 7 | baibd 538 | . . 3 ⊢ (((𝑁‘𝐹) ∈ ℝ* ∧ -∞ < (𝑁‘𝐹)) → ((𝑁‘𝐹) ∈ ℝ ↔ (𝑁‘𝐹) < +∞)) |
9 | 3, 6, 8 | syl2anc 582 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) ∈ ℝ ↔ (𝑁‘𝐹) < +∞)) |
10 | 2, 9 | bitrd 278 | 1 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) < +∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5152 ‘cfv 6553 (class class class)co 7426 ℝcr 11145 0cc0 11146 +∞cpnf 11283 -∞cmnf 11284 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 GrpHom cghm 19174 NrmGrpcngp 24506 normOp cnmo 24642 NGHom cnghm 24643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-ico 13370 df-nmo 24645 df-nghm 24646 |
This theorem is referenced by: (None) |
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