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Theorem tngval 24018
Description: Value of the function which augments a given structure 𝐺 with a norm 𝑁. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
tngval.t 𝑇 = (𝐺 toNrmGrp 𝑁)
tngval.m βˆ’ = (-gβ€˜πΊ)
tngval.d 𝐷 = (𝑁 ∘ βˆ’ )
tngval.j 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
tngval ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ π‘Š) β†’ 𝑇 = ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩))

Proof of Theorem tngval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tngval.t . 2 𝑇 = (𝐺 toNrmGrp 𝑁)
2 elex 3465 . . 3 (𝐺 ∈ 𝑉 β†’ 𝐺 ∈ V)
3 elex 3465 . . 3 (𝑁 ∈ π‘Š β†’ 𝑁 ∈ V)
4 simpl 484 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ 𝑔 = 𝐺)
5 simpr 486 . . . . . . . . 9 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ 𝑓 = 𝑁)
64fveq2d 6850 . . . . . . . . . 10 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (-gβ€˜π‘”) = (-gβ€˜πΊ))
7 tngval.m . . . . . . . . . 10 βˆ’ = (-gβ€˜πΊ)
86, 7eqtr4di 2791 . . . . . . . . 9 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (-gβ€˜π‘”) = βˆ’ )
95, 8coeq12d 5824 . . . . . . . 8 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (𝑓 ∘ (-gβ€˜π‘”)) = (𝑁 ∘ βˆ’ ))
10 tngval.d . . . . . . . 8 𝐷 = (𝑁 ∘ βˆ’ )
119, 10eqtr4di 2791 . . . . . . 7 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (𝑓 ∘ (-gβ€˜π‘”)) = 𝐷)
1211opeq2d 4841 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ ⟨(distβ€˜ndx), (𝑓 ∘ (-gβ€˜π‘”))⟩ = ⟨(distβ€˜ndx), 𝐷⟩)
134, 12oveq12d 7379 . . . . 5 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (𝑔 sSet ⟨(distβ€˜ndx), (𝑓 ∘ (-gβ€˜π‘”))⟩) = (𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩))
1411fveq2d 6850 . . . . . . 7 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (MetOpenβ€˜(𝑓 ∘ (-gβ€˜π‘”))) = (MetOpenβ€˜π·))
15 tngval.j . . . . . . 7 𝐽 = (MetOpenβ€˜π·)
1614, 15eqtr4di 2791 . . . . . 6 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ (MetOpenβ€˜(𝑓 ∘ (-gβ€˜π‘”))) = 𝐽)
1716opeq2d 4841 . . . . 5 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑓 ∘ (-gβ€˜π‘”)))⟩ = ⟨(TopSetβ€˜ndx), 𝐽⟩)
1813, 17oveq12d 7379 . . . 4 ((𝑔 = 𝐺 ∧ 𝑓 = 𝑁) β†’ ((𝑔 sSet ⟨(distβ€˜ndx), (𝑓 ∘ (-gβ€˜π‘”))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑓 ∘ (-gβ€˜π‘”)))⟩) = ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩))
19 df-tng 23963 . . . 4 toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(distβ€˜ndx), (𝑓 ∘ (-gβ€˜π‘”))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑓 ∘ (-gβ€˜π‘”)))⟩))
20 ovex 7394 . . . 4 ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩) ∈ V
2118, 19, 20ovmpoa 7514 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ V) β†’ (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩))
222, 3, 21syl2an 597 . 2 ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ π‘Š) β†’ (𝐺 toNrmGrp 𝑁) = ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩))
231, 22eqtrid 2785 1 ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ π‘Š) β†’ 𝑇 = ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βŸ¨cop 4596   ∘ ccom 5641  β€˜cfv 6500  (class class class)co 7361   sSet csts 17043  ndxcnx 17073  TopSetcts 17147  distcds 17150  -gcsg 18758  MetOpencmopn 20809   toNrmGrp ctng 23957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-tng 23963
This theorem is referenced by:  tnglem  24019  tnglemOLD  24020  tngds  24034  tngdsOLD  24035  tngtset  24036
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