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Theorem tngnm 22265
 Description: The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
tngnm.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngnm.x 𝑋 = (Base‘𝐺)
tngnm.a 𝐴 ∈ V
Assertion
Ref Expression
tngnm ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))

Proof of Theorem tngnm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 476 . . 3 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁:𝑋𝐴)
21feqmptd 6159 . 2 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (𝑥𝑋 ↦ (𝑁𝑥)))
3 tngnm.x . . . . . . . 8 𝑋 = (Base‘𝐺)
4 eqid 2610 . . . . . . . 8 (-g𝐺) = (-g𝐺)
53, 4grpsubf 17317 . . . . . . 7 (𝐺 ∈ Grp → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
65ad2antrr 758 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (-g𝐺):(𝑋 × 𝑋)⟶𝑋)
7 simpr 476 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → 𝑥𝑋)
8 eqid 2610 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
93, 8grpidcl 17273 . . . . . . . 8 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝑋)
109ad2antrr 758 . . . . . . 7 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (0g𝐺) ∈ 𝑋)
11 opelxpi 5072 . . . . . . 7 ((𝑥𝑋 ∧ (0g𝐺) ∈ 𝑋) → ⟨𝑥, (0g𝐺)⟩ ∈ (𝑋 × 𝑋))
127, 10, 11syl2anc 691 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → ⟨𝑥, (0g𝐺)⟩ ∈ (𝑋 × 𝑋))
13 fvco3 6185 . . . . . 6 (((-g𝐺):(𝑋 × 𝑋)⟶𝑋 ∧ ⟨𝑥, (0g𝐺)⟩ ∈ (𝑋 × 𝑋)) → ((𝑁 ∘ (-g𝐺))‘⟨𝑥, (0g𝐺)⟩) = (𝑁‘((-g𝐺)‘⟨𝑥, (0g𝐺)⟩)))
146, 12, 13syl2anc 691 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → ((𝑁 ∘ (-g𝐺))‘⟨𝑥, (0g𝐺)⟩) = (𝑁‘((-g𝐺)‘⟨𝑥, (0g𝐺)⟩)))
15 df-ov 6552 . . . . 5 (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)) = ((𝑁 ∘ (-g𝐺))‘⟨𝑥, (0g𝐺)⟩)
16 df-ov 6552 . . . . . 6 (𝑥(-g𝐺)(0g𝐺)) = ((-g𝐺)‘⟨𝑥, (0g𝐺)⟩)
1716fveq2i 6106 . . . . 5 (𝑁‘(𝑥(-g𝐺)(0g𝐺))) = (𝑁‘((-g𝐺)‘⟨𝑥, (0g𝐺)⟩))
1814, 15, 173eqtr4g 2669 . . . 4 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)) = (𝑁‘(𝑥(-g𝐺)(0g𝐺))))
193, 8, 4grpsubid1 17323 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝑋) → (𝑥(-g𝐺)(0g𝐺)) = 𝑥)
2019adantlr 747 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑥(-g𝐺)(0g𝐺)) = 𝑥)
2120fveq2d 6107 . . . 4 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑁‘(𝑥(-g𝐺)(0g𝐺))) = (𝑁𝑥))
2218, 21eqtr2d 2645 . . 3 (((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) ∧ 𝑥𝑋) → (𝑁𝑥) = (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)))
2322mpteq2dva 4672 . 2 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥𝑋 ↦ (𝑁𝑥)) = (𝑥𝑋 ↦ (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺))))
24 fvex 6113 . . . . . . . 8 (Base‘𝐺) ∈ V
253, 24eqeltri 2684 . . . . . . 7 𝑋 ∈ V
26 tngnm.a . . . . . . 7 𝐴 ∈ V
27 fex2 7014 . . . . . . 7 ((𝑁:𝑋𝐴𝑋 ∈ V ∧ 𝐴 ∈ V) → 𝑁 ∈ V)
2825, 26, 27mp3an23 1408 . . . . . 6 (𝑁:𝑋𝐴𝑁 ∈ V)
2928adantl 481 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 ∈ V)
30 tngnm.t . . . . . 6 𝑇 = (𝐺 toNrmGrp 𝑁)
3130, 3tngbas 22255 . . . . 5 (𝑁 ∈ V → 𝑋 = (Base‘𝑇))
3229, 31syl 17 . . . 4 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑋 = (Base‘𝑇))
3330, 4tngds 22262 . . . . . 6 (𝑁 ∈ V → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
3429, 33syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑁 ∘ (-g𝐺)) = (dist‘𝑇))
35 eqidd 2611 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑥 = 𝑥)
3630, 8tng0 22257 . . . . . 6 (𝑁 ∈ V → (0g𝐺) = (0g𝑇))
3729, 36syl 17 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (0g𝐺) = (0g𝑇))
3834, 35, 37oveq123d 6570 . . . 4 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺)) = (𝑥(dist‘𝑇)(0g𝑇)))
3932, 38mpteq12dv 4663 . . 3 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥𝑋 ↦ (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺))) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g𝑇))))
40 eqid 2610 . . . 4 (norm‘𝑇) = (norm‘𝑇)
41 eqid 2610 . . . 4 (Base‘𝑇) = (Base‘𝑇)
42 eqid 2610 . . . 4 (0g𝑇) = (0g𝑇)
43 eqid 2610 . . . 4 (dist‘𝑇) = (dist‘𝑇)
4440, 41, 42, 43nmfval 22203 . . 3 (norm‘𝑇) = (𝑥 ∈ (Base‘𝑇) ↦ (𝑥(dist‘𝑇)(0g𝑇)))
4539, 44syl6eqr 2662 . 2 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → (𝑥𝑋 ↦ (𝑥(𝑁 ∘ (-g𝐺))(0g𝐺))) = (norm‘𝑇))
462, 23, 453eqtrd 2648 1 ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  0gc0g 15923  Grpcgrp 17245  -gcsg 17247  normcnm 22191   toNrmGrp ctng 22193 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-tset 15787  df-ds 15791  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-nm 22197  df-tng 22199 This theorem is referenced by:  tngngp2  22266  tngngp  22268  tngngp3  22270  nrmtngnrm  22272  tngnrg  22288  tchnmfval  22835  tchcph  22844
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