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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 7440 | . . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇)) | |
| 2 | nmofval.1 | . . . . . 6 ⊢ 𝑁 = (𝑆 normOp 𝑇) | |
| 3 | 1, 2 | eqtr4di 2795 | . . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁) |
| 4 | 3 | cnveqd 5886 | . . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ◡(𝑠 normOp 𝑡) = ◡𝑁) |
| 5 | 4 | imaeq1d 6077 | . . 3 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (◡(𝑠 normOp 𝑡) “ ℝ) = (◡𝑁 “ ℝ)) |
| 6 | df-nghm 24730 | . . 3 ⊢ NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (◡(𝑠 normOp 𝑡) “ ℝ)) | |
| 7 | 2 | ovexi 7465 | . . . . 5 ⊢ 𝑁 ∈ V |
| 8 | 7 | cnvex 7947 | . . . 4 ⊢ ◡𝑁 ∈ V |
| 9 | 8 | imaex 7936 | . . 3 ⊢ (◡𝑁 “ ℝ) ∈ V |
| 10 | 5, 6, 9 | ovmpoa 7588 | . 2 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)) |
| 11 | 6 | mpondm0 7673 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅) |
| 12 | nmoffn 24732 | . . . . . . . . . 10 ⊢ normOp Fn (NrmGrp × NrmGrp) | |
| 13 | 12 | fndmi 6672 | . . . . . . . . 9 ⊢ dom normOp = (NrmGrp × NrmGrp) |
| 14 | 13 | ndmov 7617 | . . . . . . . 8 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅) |
| 15 | 2, 14 | eqtrid 2789 | . . . . . . 7 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅) |
| 16 | 15 | cnveqd 5886 | . . . . . 6 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ◡∅) |
| 17 | cnv0 6160 | . . . . . 6 ⊢ ◡∅ = ∅ | |
| 18 | 16, 17 | eqtrdi 2793 | . . . . 5 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → ◡𝑁 = ∅) |
| 19 | 18 | imaeq1d 6077 | . . . 4 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = (∅ “ ℝ)) |
| 20 | 0ima 6096 | . . . 4 ⊢ (∅ “ ℝ) = ∅ | |
| 21 | 19, 20 | eqtrdi 2793 | . . 3 ⊢ (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (◡𝑁 “ ℝ) = ∅) |
| 22 | 11, 21 | eqtr4d 2780 | . 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 395 = wceq 1540 ∈ wcel 2108 ∅c0 4333 × cxp 5683 ◡ccnv 5684 “ cima 5688 (class class class)co 7431 ℝcr 11154 NrmGrpcngp 24590 normOp cnmo 24726 NGHom cnghm 24727 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-ico 13393 df-nmo 24729 df-nghm 24730 |
| This theorem is referenced by: isnghm 24744 |
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