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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  ℝcr 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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