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Mirrors > Home > MPE Home > Th. List > nghmfval | 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 |
---|---|
nghmfval | ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 7428 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇)) | |
2 | nmofval.1 | . . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
3 | 1, 2 | eqtr4di 2783 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁) |
4 | 3 | cnveqd 5878 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ◡(𝑠 normOp 𝑡) = ◡𝑁) |
5 | 4 | imaeq1d 6063 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (◡(𝑠 normOp 𝑡) “ ℝ) = (◡𝑁 “ ℝ)) |
6 | df-nghm 24670 | . . 3 ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) | |
7 | 2 | ovexi 7453 | . . . . 5 ⊢ 𝑁 ∈ V |
8 | 7 | cnvex 7933 | . . . 4 ⊢ ◡𝑁 ∈ V |
9 | 8 | imaex 7922 | . . 3 ⊢ (◡𝑁 “ ℝ) ∈ V |
10 | 5, 6, 9 | ovmpoa 7576 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
11 | 6 | mpondm0 7661 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅) |
12 | nmoffn 24672 | . . . . . . . . . 10 ⊢ normOp Fn (NrmGrp × NrmGrp) | |
13 | 12 | fndmi 6659 | . . . . . . . . 9 ⊢ dom normOp = (NrmGrp × NrmGrp) |
14 | 13 | ndmov 7605 | . . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅) |
15 | 2, 14 | eqtrid 2777 | . . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅) |
16 | 15 | cnveqd 5878 | . . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅) |
17 | cnv0 6147 | . . . . . 6 ⊢ ◡∅ = ∅ | |
18 | 16, 17 | eqtrdi 2781 | . . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅) |
19 | 18 | imaeq1d 6063 | . . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ)) |
20 | 0ima 6082 | . . . 4 ⊢ (∅ “ ℝ) = ∅ | |
21 | 19, 20 | eqtrdi 2781 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅) |
22 | 11, 21 | eqtr4d 2768 | . 2 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
23 | 10, 22 | pm2.61i 182 | 1 ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∅c0 4322 × cxp 5676 ◡ccnv 5677 “ cima 5681 (class class class)co 7419 ℝcr 11139 NrmGrpcngp 24530 normOp cnmo 24666 NGHom cnghm 24667 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-ico 13365 df-nmo 24669 df-nghm 24670 |
This theorem is referenced by: isnghm 24684 |
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