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Theorem nghmfval 23024
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 6979 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 normOp 𝑡) = (𝑆 normOp 𝑇))
2 nmofval.1 . . . . . 6 𝑁 = (𝑆 normOp 𝑇)
31, 2syl6eqr 2826 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁)
43cnveqd 5589 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑠 normOp 𝑡) = 𝑁)
54imaeq1d 5763 . . 3 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑠 normOp 𝑡) “ ℝ) = (𝑁 “ ℝ))
6 df-nghm 23011 . . 3 NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ ((𝑠 normOp 𝑡) “ ℝ))
72ovexi 7003 . . . . 5 𝑁 ∈ V
87cnvex 7439 . . . 4 𝑁 ∈ V
98imaex 7430 . . 3 (𝑁 “ ℝ) ∈ V
105, 6, 9ovmpoa 7115 . 2 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (𝑁 “ ℝ))
116mpondm0 7199 . . 3 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = ∅)
12 nmoffn 23013 . . . . . . . . . 10 normOp Fn (NrmGrp × NrmGrp)
13 fndm 6282 . . . . . . . . . 10 ( normOp Fn (NrmGrp × NrmGrp) → dom normOp = (NrmGrp × NrmGrp))
1412, 13ax-mp 5 . . . . . . . . 9 dom normOp = (NrmGrp × NrmGrp)
1514ndmov 7142 . . . . . . . 8 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 normOp 𝑇) = ∅)
162, 15syl5eq 2820 . . . . . . 7 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
1716cnveqd 5589 . . . . . 6 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
18 cnv0 5833 . . . . . 6 ∅ = ∅
1917, 18syl6eq 2824 . . . . 5 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = ∅)
2019imaeq1d 5763 . . . 4 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁 “ ℝ) = (∅ “ ℝ))
21 0ima 5780 . . . 4 (∅ “ ℝ) = ∅
2220, 21syl6eq 2824 . . 3 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁 “ ℝ) = ∅)
2311, 22eqtr4d 2811 . 2 (¬ (𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑆 NGHom 𝑇) = (𝑁 “ ℝ))
2410, 23pm2.61i 177 1 (𝑆 NGHom 𝑇) = (𝑁 “ ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 387   = wceq 1507  wcel 2048  c0 4173   × cxp 5398  ccnv 5399  dom cdm 5400  cima 5403   Fn wfn 6177  (class class class)co 6970  cr 10326  NrmGrpcngp 22880   normOp cnmo 23007   NGHom cnghm 23008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404  ax-pre-sup 10405
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-po 5319  df-so 5320  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-sup 8693  df-inf 8694  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-ico 12553  df-nmo 23010  df-nghm 23011
This theorem is referenced by:  isnghm  23025
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