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Theorem nghmfval 22805
Description: A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
Hypothesis
Ref Expression
nmofval.1 𝑁 = (𝑆 normOp 𝑇)
Assertion
Ref Expression
nghmfval (𝑆 NGHom 𝑇) = (𝑁 “ ℝ)

Proof of Theorem nghmfval
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq12 6851 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇))
2 nmofval.1 . . . . . 6 𝑁 = (𝑆 normOp 𝑇)
31, 2syl6eqr 2817 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁)
43cnveqd 5466 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁)
54imaeq1d 5647 . . 3 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑠 normOp 𝑡) “ ℝ) = (𝑁 “ ℝ))
6 df-nghm 22792 . . 3 NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ ((𝑠 normOp 𝑡) “ ℝ))
72ovexi 6875 . . . . 5 𝑁 ∈ V
87cnvex 7311 . . . 4 𝑁 ∈ V
98imaex 7302 . . 3 (𝑁 “ ℝ) ∈ V
105, 6, 9ovmpt2a 6989 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (𝑁 “ ℝ))
116mpt2ndm0 7073 . . 3 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅)
12 nmoffn 22794 . . . . . . . . . 10 normOp Fn (NrmGrp × NrmGrp)
13 fndm 6168 . . . . . . . . . 10 ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp))
1412, 13ax-mp 5 . . . . . . . . 9 dom normOp = (NrmGrp × NrmGrp)
1514ndmov 7016 . . . . . . . 8 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅)
162, 15syl5eq 2811 . . . . . . 7 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
1716cnveqd 5466 . . . . . 6 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
18 cnv0 5718 . . . . . 6 ∅ = ∅
1917, 18syl6eq 2815 . . . . 5 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
2019imaeq1d 5647 . . . 4 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁 “ ℝ) = (∅ “ ℝ))
21 0ima 5664 . . . 4 (∅ “ ℝ) = ∅
2220, 21syl6eq 2815 . . 3 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁 “ ℝ) = ∅)
2311, 22eqtr4d 2802 . 2 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (𝑁 “ ℝ))
2410, 23pm2.61i 176 1 (𝑆 NGHom 𝑇) = (𝑁 “ ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1652  wcel 2155  c0 4079   × cxp 5275  ccnv 5276  dom cdm 5277  cima 5280   Fn wfn 6063  (class class class)co 6842  cr 10188  NrmGrpcngp 22661   normOp cnmo 22788   NGHom cnghm 22789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-ico 12383  df-nmo 22791  df-nghm 22792
This theorem is referenced by:  isnghm  22806
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