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Theorem elntg 28711
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 28160 . . 3 LineG = Slot (LineGβ€˜ndx)
2 fvex 6894 . . . 4 (EEGβ€˜π‘) ∈ V
32a1i 11 . . 3 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) ∈ V)
4 eengstr 28707 . . . . 5 (𝑁 ∈ β„• β†’ (EEGβ€˜π‘) Struct ⟨1, 17⟩)
5 structn0fun 17083 . . . . 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 17085 . . . . . 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 6560 . . . 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 5454 . . . . . 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 4759 . . . . 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 4169 . . . . 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 28706 . . . 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 2832 . . 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 6894 . . . . 5 (π”Όβ€˜π‘) ∈ V
1817difexi 5318 . . . . 5 ((π”Όβ€˜π‘) βˆ– {π‘₯}) ∈ V
1917, 18mpoex 8059 . . . 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 17134 . 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 28708 . . . 4 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = (Baseβ€˜(EEGβ€˜π‘)))
23 elntg.1 . . . 4 𝑃 = (Baseβ€˜(EEGβ€˜π‘))
2422, 23eqtr4di 2782 . . 3 (𝑁 ∈ β„• β†’ (π”Όβ€˜π‘) = 𝑃)
2524difeq1d 4113 . . . 4 (𝑁 ∈ β„• β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2625adantr 480 . . 3 ((𝑁 ∈ β„• ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((π”Όβ€˜π‘) βˆ– {π‘₯}) = (𝑃 βˆ– {π‘₯}))
2724adantr 480 . . . 4 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) β†’ (π”Όβ€˜π‘) = 𝑃)
28 simpll 764 . . . . . 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 774 . . . . . . 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 2827 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ 𝑃)
33 simplrr 775 . . . . . . . 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 3952 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
3534, 31eleqtrd 2827 . . . . . 6 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ 𝑃)
36 simpr 484 . . . . . . 7 (((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ ((π”Όβ€˜π‘) βˆ– {π‘₯}))) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
3736, 31eleqtrd 2827 . . . . . 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 28709 . . . . 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 28709 . . . . 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 28709 . . . . 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 1431 . . . 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 3440 . . 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 7475 . 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 2766 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 395   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  {crab 3424  Vcvv 3466   βˆ– cdif 3937   βˆͺ cun 3938  βˆ…c0 4314  {csn 4620  {cpr 4622  βŸ¨cop 4626   class class class wbr 5138  β—‘ccnv 5665  Fun wfun 6527  β€˜cfv 6533  (class class class)co 7401   ∈ cmpo 7403  1c1 11107   βˆ’ cmin 11441  β„•cn 12209  2c2 12264  7c7 12269  cdc 12674  ...cfz 13481  β†‘cexp 14024  Ξ£csu 15629   Struct cstr 17078  ndxcnx 17125  Basecbs 17143  distcds 17205  Itvcitv 28153  LineGclng 28154  π”Όcee 28615   Btwn cbtwn 28616  EEGceeng 28704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-seq 13964  df-sum 15630  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-ds 17218  df-itv 28155  df-lng 28156  df-eeng 28705
This theorem is referenced by:  elntg2  28712  eengtrkg  28713
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