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Theorem elntg 27982
Description: The line definition in the Tarski structure for the Euclidean geometry. (Contributed by Thierry Arnoux, 7-Apr-2019.)
Hypotheses
Ref Expression
elntg.1 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
elntg.2 𝐼 = (Itvβ€˜(EEGβ€˜π‘))
Assertion
Ref Expression
elntg (𝑁 ∈ β„• β†’ (LineGβ€˜(EEGβ€˜π‘)) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
Distinct variable groups:   π‘₯,𝑦,𝑧,𝑁   𝑧,𝑃
Allowed substitution hints:   𝑃(π‘₯,𝑦)   𝐼(π‘₯,𝑦,𝑧)

Proof of Theorem elntg
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 lngid 27431 . . 3 LineG = Slot (LineGβ€˜ndx)
2 fvex 6859 . . . 4 (EEGβ€˜π‘) ∈ V
32a1i 11 . . 3 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ V)
4 eengstr 27978 . . . . 5 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) Struct ⟨1, 17⟩)
5 structn0fun 17031 . . . . 5 ((EEGβ€˜π‘) Struct ⟨1, 17⟩ β†’ Fun ((EEGβ€˜π‘) βˆ– {βˆ…}))
64, 5syl 17 . . . 4 (𝑁 ∈ β„• β†’ Fun ((EEGβ€˜π‘) βˆ– {βˆ…}))
7 structcnvcnv 17033 . . . . . 6 ((EEGβ€˜π‘) Struct ⟨1, 17⟩ β†’ β—‘β—‘(EEGβ€˜π‘) = ((EEGβ€˜π‘) βˆ– {βˆ…}))
84, 7syl 17 . . . . 5 (𝑁 ∈ β„• β†’ β—‘β—‘(EEGβ€˜π‘) = ((EEGβ€˜π‘) βˆ– {βˆ…}))
98funeqd 6527 . . . 4 (𝑁 ∈ β„• β†’ (Fun β—‘β—‘(EEGβ€˜π‘) ↔ Fun ((EEGβ€˜π‘) βˆ– {βˆ…})))
106, 9mpbird 257 . . 3 (𝑁 ∈ β„• β†’ Fun β—‘β—‘(EEGβ€˜π‘))
11 opex 5425 . . . . . 6 ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ V
1211prid2 4728 . . . . 5 ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}
13 elun2 4141 . . . . 5 (⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩} β†’ ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
1412, 13ax-mp 5 . . . 4 ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩})
15 eengv 27977 . . . 4 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) = ({⟨(Baseβ€˜ndx), (π”Όβ€˜π‘)⟩, ⟨(distβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ Σ𝑖 ∈ (1...𝑁)(((π‘₯β€˜π‘–) βˆ’ (π‘¦β€˜π‘–))↑2))⟩} βˆͺ {⟨(Itvβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ (π”Όβ€˜π‘) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ 𝑧 Btwn ⟨π‘₯, π‘¦βŸ©})⟩, ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩}))
1614, 15eleqtrrid 2841 . . 3 (𝑁 ∈ β„• β†’ ⟨(LineGβ€˜ndx), (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)})⟩ ∈ (EEGβ€˜π‘))
17 fvex 6859 . . . . 5 (π”Όβ€˜π‘) ∈ V
1817difexi 5289 . . . . 5 ((π”Όβ€˜π‘) βˆ– {π‘₯}) ∈ V
1917, 18mpoex 8016 . . . 4 (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) ∈ V
2019a1i 11 . . 3 (𝑁 ∈ β„• β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) ∈ V)
211, 3, 10, 16, 20strfv2d 17082 . 2 (𝑁 ∈ β„• β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) = (LineGβ€˜(EEGβ€˜π‘)))
22 eengbas 27979 . . . 4 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
23 elntg.1 . . . 4 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
2422, 23eqtr4di 2791 . . 3 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = 𝑃)
2524difeq1d 4085 . . . 4 (𝑁 ∈ β„• β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2625adantr 482 . . 3 ((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2724adantr 482 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) β†’ (π”Όβ€˜π‘) = 𝑃)
28 simpll 766 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
29 elntg.2 . . . . . 6 𝐼 = (Itvβ€˜(EEGβ€˜π‘))
30 simplrl 776 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
3128, 24syl 17 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (π”Όβ€˜π‘) = 𝑃)
3230, 31eleqtrd 2836 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ 𝑃)
33 simplrr 777 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))
3433eldifad 3926 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
3534, 31eleqtrd 2836 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ 𝑃)
36 simpr 486 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
3736, 31eleqtrd 2836 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ 𝑃)
3828, 23, 29, 32, 35, 37ebtwntg 27980 . . . . 5 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ↔ 𝑧 ∈ (π‘₯𝐼𝑦)))
3928, 23, 29, 37, 35, 32ebtwntg 27980 . . . . 5 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ↔ π‘₯ ∈ (𝑧𝐼𝑦)))
4028, 23, 29, 32, 37, 35ebtwntg 27980 . . . . 5 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (𝑦 Btwn ⟨π‘₯, π‘§βŸ© ↔ 𝑦 ∈ (π‘₯𝐼𝑧)))
4138, 39, 403orbi123d 1436 . . . 4 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ ((𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©) ↔ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))))
4227, 41rabeqbidva 3422 . . 3 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) β†’ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)} = {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))})
4324, 26, 42mpoeq123dva 7435 . 2 (𝑁 ∈ β„• β†’ (π‘₯ ∈ (π”Όβ€˜π‘), 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}) ↦ {𝑧 ∈ (π”Όβ€˜π‘) ∣ (𝑧 Btwn ⟨π‘₯, π‘¦βŸ© ∨ π‘₯ Btwn βŸ¨π‘§, π‘¦βŸ© ∨ 𝑦 Btwn ⟨π‘₯, π‘§βŸ©)}) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
4421, 43eqtr3d 2775 1 (𝑁 ∈ β„• β†’ (LineGβ€˜(EEGβ€˜π‘)) = (π‘₯ ∈ 𝑃, 𝑦 ∈ (𝑃 βˆ– {π‘₯}) ↦ {𝑧 ∈ 𝑃 ∣ (𝑧 ∈ (π‘₯𝐼𝑦) ∨ π‘₯ ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (π‘₯𝐼𝑧))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ w3o 1087   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3447   βˆ– cdif 3911   βˆͺ cun 3912  βˆ…c0 4286  {csn 4590  {cpr 4592  βŸ¨cop 4596   class class class wbr 5109  β—‘ccnv 5636  Fun wfun 6494  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  1c1 11060   βˆ’ cmin 11393  β„•cn 12161  2c2 12216  7c7 12221  cdc 12626  ...cfz 13433  β†‘cexp 13976  Ξ£csu 15579   Struct cstr 17026  ndxcnx 17073  Basecbs 17091  distcds 17150  Itvcitv 27424  LineGclng 27425  π”Όcee 27886   Btwn cbtwn 27887  EEGceeng 27975
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-uz 12772  df-fz 13434  df-seq 13916  df-sum 15580  df-struct 17027  df-slot 17062  df-ndx 17074  df-base 17092  df-ds 17163  df-itv 27426  df-lng 27427  df-eeng 27976
This theorem is referenced by:  elntg2  27983  eengtrkg  27984
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