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Theorem axlowdimlem17 28206
Description: Lemma for axlowdim 28209. Establish a congruence result. (Contributed by Scott Fenton, 22-Apr-2013.) (Proof shortened by Mario Carneiro, 22-May-2014.)
Hypotheses
Ref Expression
axlowdimlem16.1 𝑃 = ({⟨3, -1⟩} βˆͺ (((1...𝑁) βˆ– {3}) Γ— {0}))
axlowdimlem16.2 𝑄 = ({⟨(𝐼 + 1), 1⟩} βˆͺ (((1...𝑁) βˆ– {(𝐼 + 1)}) Γ— {0}))
axlowdimlem17.3 𝐴 = ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0}))
axlowdimlem17.4 𝑋 ∈ ℝ
axlowdimlem17.5 π‘Œ ∈ ℝ
Assertion
Ref Expression
axlowdimlem17 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ βŸ¨π‘ƒ, 𝐴⟩CgrβŸ¨π‘„, 𝐴⟩)

Proof of Theorem axlowdimlem17
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 uzuzle23 12870 . . . . . . . . . . . 12 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
21ad2antrr 725 . . . . . . . . . . 11 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
3 fzss2 13538 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ (1...2) βŠ† (1...𝑁))
42, 3syl 17 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (1...2) βŠ† (1...𝑁))
5 simpr 486 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ 𝑖 ∈ (1...2))
64, 5sseldd 3983 . . . . . . . . 9 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ 𝑖 ∈ (1...𝑁))
7 fznuz 13580 . . . . . . . . . . 11 (𝑖 ∈ (1...2) β†’ Β¬ 𝑖 ∈ (β„€β‰₯β€˜(2 + 1)))
87adantl 483 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ Β¬ 𝑖 ∈ (β„€β‰₯β€˜(2 + 1)))
9 3z 12592 . . . . . . . . . . . . . 14 3 ∈ β„€
10 uzid 12834 . . . . . . . . . . . . . 14 (3 ∈ β„€ β†’ 3 ∈ (β„€β‰₯β€˜3))
119, 10ax-mp 5 . . . . . . . . . . . . 13 3 ∈ (β„€β‰₯β€˜3)
12 df-3 12273 . . . . . . . . . . . . . 14 3 = (2 + 1)
1312fveq2i 6892 . . . . . . . . . . . . 13 (β„€β‰₯β€˜3) = (β„€β‰₯β€˜(2 + 1))
1411, 13eleqtri 2832 . . . . . . . . . . . 12 3 ∈ (β„€β‰₯β€˜(2 + 1))
15 eleq1 2822 . . . . . . . . . . . 12 (𝑖 = 3 β†’ (𝑖 ∈ (β„€β‰₯β€˜(2 + 1)) ↔ 3 ∈ (β„€β‰₯β€˜(2 + 1))))
1614, 15mpbiri 258 . . . . . . . . . . 11 (𝑖 = 3 β†’ 𝑖 ∈ (β„€β‰₯β€˜(2 + 1)))
1716necon3bi 2968 . . . . . . . . . 10 (Β¬ 𝑖 ∈ (β„€β‰₯β€˜(2 + 1)) β†’ 𝑖 β‰  3)
188, 17syl 17 . . . . . . . . 9 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ 𝑖 β‰  3)
19 axlowdimlem16.1 . . . . . . . . . 10 𝑃 = ({⟨3, -1⟩} βˆͺ (((1...𝑁) βˆ– {3}) Γ— {0}))
2019axlowdimlem9 28198 . . . . . . . . 9 ((𝑖 ∈ (1...𝑁) ∧ 𝑖 β‰  3) β†’ (π‘ƒβ€˜π‘–) = 0)
216, 18, 20syl2anc 585 . . . . . . . 8 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (π‘ƒβ€˜π‘–) = 0)
22 elfzuz 13494 . . . . . . . . . . . . . 14 (𝐼 ∈ (2...(𝑁 βˆ’ 1)) β†’ 𝐼 ∈ (β„€β‰₯β€˜2))
2322ad2antlr 726 . . . . . . . . . . . . 13 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ 𝐼 ∈ (β„€β‰₯β€˜2))
24 eluzp1p1 12847 . . . . . . . . . . . . 13 (𝐼 ∈ (β„€β‰₯β€˜2) β†’ (𝐼 + 1) ∈ (β„€β‰₯β€˜(2 + 1)))
2523, 24syl 17 . . . . . . . . . . . 12 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (𝐼 + 1) ∈ (β„€β‰₯β€˜(2 + 1)))
26 uzss 12842 . . . . . . . . . . . 12 ((𝐼 + 1) ∈ (β„€β‰₯β€˜(2 + 1)) β†’ (β„€β‰₯β€˜(𝐼 + 1)) βŠ† (β„€β‰₯β€˜(2 + 1)))
2725, 26syl 17 . . . . . . . . . . 11 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (β„€β‰₯β€˜(𝐼 + 1)) βŠ† (β„€β‰₯β€˜(2 + 1)))
2827, 8ssneldd 3985 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ Β¬ 𝑖 ∈ (β„€β‰₯β€˜(𝐼 + 1)))
29 eluzelz 12829 . . . . . . . . . . . . . 14 ((𝐼 + 1) ∈ (β„€β‰₯β€˜(2 + 1)) β†’ (𝐼 + 1) ∈ β„€)
3025, 29syl 17 . . . . . . . . . . . . 13 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (𝐼 + 1) ∈ β„€)
31 uzid 12834 . . . . . . . . . . . . 13 ((𝐼 + 1) ∈ β„€ β†’ (𝐼 + 1) ∈ (β„€β‰₯β€˜(𝐼 + 1)))
3230, 31syl 17 . . . . . . . . . . . 12 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (𝐼 + 1) ∈ (β„€β‰₯β€˜(𝐼 + 1)))
33 eleq1 2822 . . . . . . . . . . . 12 (𝑖 = (𝐼 + 1) β†’ (𝑖 ∈ (β„€β‰₯β€˜(𝐼 + 1)) ↔ (𝐼 + 1) ∈ (β„€β‰₯β€˜(𝐼 + 1))))
3432, 33syl5ibrcom 246 . . . . . . . . . . 11 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (𝑖 = (𝐼 + 1) β†’ 𝑖 ∈ (β„€β‰₯β€˜(𝐼 + 1))))
3534necon3bd 2955 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (Β¬ 𝑖 ∈ (β„€β‰₯β€˜(𝐼 + 1)) β†’ 𝑖 β‰  (𝐼 + 1)))
3628, 35mpd 15 . . . . . . . . 9 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ 𝑖 β‰  (𝐼 + 1))
37 axlowdimlem16.2 . . . . . . . . . 10 𝑄 = ({⟨(𝐼 + 1), 1⟩} βˆͺ (((1...𝑁) βˆ– {(𝐼 + 1)}) Γ— {0}))
3837axlowdimlem12 28201 . . . . . . . . 9 ((𝑖 ∈ (1...𝑁) ∧ 𝑖 β‰  (𝐼 + 1)) β†’ (π‘„β€˜π‘–) = 0)
396, 36, 38syl2anc 585 . . . . . . . 8 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (π‘„β€˜π‘–) = 0)
4021, 39eqtr4d 2776 . . . . . . 7 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (π‘ƒβ€˜π‘–) = (π‘„β€˜π‘–))
4140oveq1d 7421 . . . . . 6 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ ((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–)) = ((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–)))
4241oveq1d 7421 . . . . 5 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...2)) β†’ (((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = (((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2))
4342sumeq2dv 15646 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (1...2)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (1...2)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2))
4419, 37axlowdimlem16 28205 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (3...𝑁)((π‘ƒβ€˜π‘–)↑2) = Σ𝑖 ∈ (3...𝑁)((π‘„β€˜π‘–)↑2))
45 axlowdimlem17.3 . . . . . . . . . . . . 13 𝐴 = ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0}))
4645fveq1i 6890 . . . . . . . . . . . 12 (π΄β€˜π‘–) = (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0}))β€˜π‘–)
47 axlowdimlem2 28191 . . . . . . . . . . . . 13 ((1...2) ∩ (3...𝑁)) = βˆ…
48 axlowdimlem17.4 . . . . . . . . . . . . . . . 16 𝑋 ∈ ℝ
49 axlowdimlem17.5 . . . . . . . . . . . . . . . 16 π‘Œ ∈ ℝ
5048, 49axlowdimlem4 28193 . . . . . . . . . . . . . . 15 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©}:(1...2)βŸΆβ„
51 ffn 6715 . . . . . . . . . . . . . . 15 ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©}:(1...2)βŸΆβ„ β†’ {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} Fn (1...2))
5250, 51ax-mp 5 . . . . . . . . . . . . . 14 {⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} Fn (1...2)
53 axlowdimlem1 28190 . . . . . . . . . . . . . . 15 ((3...𝑁) Γ— {0}):(3...𝑁)βŸΆβ„
54 ffn 6715 . . . . . . . . . . . . . . 15 (((3...𝑁) Γ— {0}):(3...𝑁)βŸΆβ„ β†’ ((3...𝑁) Γ— {0}) Fn (3...𝑁))
5553, 54ax-mp 5 . . . . . . . . . . . . . 14 ((3...𝑁) Γ— {0}) Fn (3...𝑁)
56 fvun2 6981 . . . . . . . . . . . . . 14 (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} Fn (1...2) ∧ ((3...𝑁) Γ— {0}) Fn (3...𝑁) ∧ (((1...2) ∩ (3...𝑁)) = βˆ… ∧ 𝑖 ∈ (3...𝑁))) β†’ (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0}))β€˜π‘–) = (((3...𝑁) Γ— {0})β€˜π‘–))
5752, 55, 56mp3an12 1452 . . . . . . . . . . . . 13 ((((1...2) ∩ (3...𝑁)) = βˆ… ∧ 𝑖 ∈ (3...𝑁)) β†’ (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0}))β€˜π‘–) = (((3...𝑁) Γ— {0})β€˜π‘–))
5847, 57mpan 689 . . . . . . . . . . . 12 (𝑖 ∈ (3...𝑁) β†’ (({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0}))β€˜π‘–) = (((3...𝑁) Γ— {0})β€˜π‘–))
5946, 58eqtrid 2785 . . . . . . . . . . 11 (𝑖 ∈ (3...𝑁) β†’ (π΄β€˜π‘–) = (((3...𝑁) Γ— {0})β€˜π‘–))
60 c0ex 11205 . . . . . . . . . . . 12 0 ∈ V
6160fvconst2 7202 . . . . . . . . . . 11 (𝑖 ∈ (3...𝑁) β†’ (((3...𝑁) Γ— {0})β€˜π‘–) = 0)
6259, 61eqtrd 2773 . . . . . . . . . 10 (𝑖 ∈ (3...𝑁) β†’ (π΄β€˜π‘–) = 0)
6362adantl 483 . . . . . . . . 9 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ (π΄β€˜π‘–) = 0)
6463oveq2d 7422 . . . . . . . 8 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ ((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–)) = ((π‘ƒβ€˜π‘–) βˆ’ 0))
6519axlowdimlem7 28196 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
6665ad2antrr 725 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
67 3nn 12288 . . . . . . . . . . . . . 14 3 ∈ β„•
68 nnuz 12862 . . . . . . . . . . . . . 14 β„• = (β„€β‰₯β€˜1)
6967, 68eleqtri 2832 . . . . . . . . . . . . 13 3 ∈ (β„€β‰₯β€˜1)
70 fzss1 13537 . . . . . . . . . . . . 13 (3 ∈ (β„€β‰₯β€˜1) β†’ (3...𝑁) βŠ† (1...𝑁))
7169, 70ax-mp 5 . . . . . . . . . . . 12 (3...𝑁) βŠ† (1...𝑁)
7271sseli 3978 . . . . . . . . . . 11 (𝑖 ∈ (3...𝑁) β†’ 𝑖 ∈ (1...𝑁))
7372adantl 483 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ 𝑖 ∈ (1...𝑁))
74 fveecn 28150 . . . . . . . . . 10 ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑖 ∈ (1...𝑁)) β†’ (π‘ƒβ€˜π‘–) ∈ β„‚)
7566, 73, 74syl2anc 585 . . . . . . . . 9 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ (π‘ƒβ€˜π‘–) ∈ β„‚)
7675subid1d 11557 . . . . . . . 8 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ ((π‘ƒβ€˜π‘–) βˆ’ 0) = (π‘ƒβ€˜π‘–))
7764, 76eqtrd 2773 . . . . . . 7 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ ((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–)) = (π‘ƒβ€˜π‘–))
7877oveq1d 7421 . . . . . 6 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ (((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = ((π‘ƒβ€˜π‘–)↑2))
7978sumeq2dv 15646 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (3...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (3...𝑁)((π‘ƒβ€˜π‘–)↑2))
8063oveq2d 7422 . . . . . . . 8 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ ((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–)) = ((π‘„β€˜π‘–) βˆ’ 0))
81 eluzge3nn 12871 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ β„•)
82 2eluzge1 12875 . . . . . . . . . . . . 13 2 ∈ (β„€β‰₯β€˜1)
83 fzss1 13537 . . . . . . . . . . . . 13 (2 ∈ (β„€β‰₯β€˜1) β†’ (2...(𝑁 βˆ’ 1)) βŠ† (1...(𝑁 βˆ’ 1)))
8482, 83ax-mp 5 . . . . . . . . . . . 12 (2...(𝑁 βˆ’ 1)) βŠ† (1...(𝑁 βˆ’ 1))
8584sseli 3978 . . . . . . . . . . 11 (𝐼 ∈ (2...(𝑁 βˆ’ 1)) β†’ 𝐼 ∈ (1...(𝑁 βˆ’ 1)))
8637axlowdimlem10 28199 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ 𝐼 ∈ (1...(𝑁 βˆ’ 1))) β†’ 𝑄 ∈ (π”Όβ€˜π‘))
8781, 85, 86syl2an 597 . . . . . . . . . 10 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ 𝑄 ∈ (π”Όβ€˜π‘))
88 fveecn 28150 . . . . . . . . . 10 ((𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑖 ∈ (1...𝑁)) β†’ (π‘„β€˜π‘–) ∈ β„‚)
8987, 72, 88syl2an 597 . . . . . . . . 9 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ (π‘„β€˜π‘–) ∈ β„‚)
9089subid1d 11557 . . . . . . . 8 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ ((π‘„β€˜π‘–) βˆ’ 0) = (π‘„β€˜π‘–))
9180, 90eqtrd 2773 . . . . . . 7 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ ((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–)) = (π‘„β€˜π‘–))
9291oveq1d 7421 . . . . . 6 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (3...𝑁)) β†’ (((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = ((π‘„β€˜π‘–)↑2))
9392sumeq2dv 15646 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (3...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (3...𝑁)((π‘„β€˜π‘–)↑2))
9444, 79, 933eqtr4d 2783 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (3...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (3...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2))
9543, 94oveq12d 7424 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (Σ𝑖 ∈ (1...2)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) + Σ𝑖 ∈ (3...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2)) = (Σ𝑖 ∈ (1...2)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) + Σ𝑖 ∈ (3...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2)))
9647a1i 11 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ ((1...2) ∩ (3...𝑁)) = βˆ…)
97 eluzelre 12830 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ ℝ)
98 eluzle 12832 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 3 ≀ 𝑁)
99 2re 12283 . . . . . . . . . . . 12 2 ∈ ℝ
100 3re 12289 . . . . . . . . . . . 12 3 ∈ ℝ
101 2lt3 12381 . . . . . . . . . . . 12 2 < 3
10299, 100, 101ltleii 11334 . . . . . . . . . . 11 2 ≀ 3
103 letr 11305 . . . . . . . . . . . 12 ((2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ ((2 ≀ 3 ∧ 3 ≀ 𝑁) β†’ 2 ≀ 𝑁))
10499, 100, 103mp3an12 1452 . . . . . . . . . . 11 (𝑁 ∈ ℝ β†’ ((2 ≀ 3 ∧ 3 ≀ 𝑁) β†’ 2 ≀ 𝑁))
105102, 104mpani 695 . . . . . . . . . 10 (𝑁 ∈ ℝ β†’ (3 ≀ 𝑁 β†’ 2 ≀ 𝑁))
10697, 98, 105sylc 65 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 2 ≀ 𝑁)
107 1le2 12418 . . . . . . . . 9 1 ≀ 2
108106, 107jctil 521 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ (1 ≀ 2 ∧ 2 ≀ 𝑁))
109108adantr 482 . . . . . . 7 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (1 ≀ 2 ∧ 2 ≀ 𝑁))
110 eluzelz 12829 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝑁 ∈ β„€)
111110adantr 482 . . . . . . . 8 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ 𝑁 ∈ β„€)
112 2z 12591 . . . . . . . . 9 2 ∈ β„€
113 1z 12589 . . . . . . . . 9 1 ∈ β„€
114 elfz 13487 . . . . . . . . 9 ((2 ∈ β„€ ∧ 1 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (2 ∈ (1...𝑁) ↔ (1 ≀ 2 ∧ 2 ≀ 𝑁)))
115112, 113, 114mp3an12 1452 . . . . . . . 8 (𝑁 ∈ β„€ β†’ (2 ∈ (1...𝑁) ↔ (1 ≀ 2 ∧ 2 ≀ 𝑁)))
116111, 115syl 17 . . . . . . 7 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (2 ∈ (1...𝑁) ↔ (1 ≀ 2 ∧ 2 ≀ 𝑁)))
117109, 116mpbird 257 . . . . . 6 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ 2 ∈ (1...𝑁))
118 fzsplit 13524 . . . . . 6 (2 ∈ (1...𝑁) β†’ (1...𝑁) = ((1...2) βˆͺ ((2 + 1)...𝑁)))
119117, 118syl 17 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (1...𝑁) = ((1...2) βˆͺ ((2 + 1)...𝑁)))
12012oveq1i 7416 . . . . . 6 (3...𝑁) = ((2 + 1)...𝑁)
121120uneq2i 4160 . . . . 5 ((1...2) βˆͺ (3...𝑁)) = ((1...2) βˆͺ ((2 + 1)...𝑁))
122119, 121eqtr4di 2791 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (1...𝑁) = ((1...2) βˆͺ (3...𝑁)))
123 fzfid 13935 . . . 4 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (1...𝑁) ∈ Fin)
12465ad2antrr 725 . . . . . . 7 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
125124, 74sylancom 589 . . . . . 6 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ (π‘ƒβ€˜π‘–) ∈ β„‚)
12648, 49axlowdimlem5 28194 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ ({⟨1, π‘‹βŸ©, ⟨2, π‘ŒβŸ©} βˆͺ ((3...𝑁) Γ— {0})) ∈ (π”Όβ€˜π‘))
12745, 126eqeltrid 2838 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
1281, 127syl 17 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜3) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
129128ad2antrr 725 . . . . . . 7 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
130 fveecn 28150 . . . . . . 7 ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑖 ∈ (1...𝑁)) β†’ (π΄β€˜π‘–) ∈ β„‚)
131129, 130sylancom 589 . . . . . 6 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ (π΄β€˜π‘–) ∈ β„‚)
132125, 131subcld 11568 . . . . 5 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ ((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–)) ∈ β„‚)
133132sqcld 14106 . . . 4 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ (((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) ∈ β„‚)
13496, 122, 123, 133fsumsplit 15684 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (1...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = (Σ𝑖 ∈ (1...2)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) + Σ𝑖 ∈ (3...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2)))
13587, 88sylan 581 . . . . . 6 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ (π‘„β€˜π‘–) ∈ β„‚)
136135, 131subcld 11568 . . . . 5 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ ((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–)) ∈ β„‚)
137136sqcld 14106 . . . 4 (((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) ∧ 𝑖 ∈ (1...𝑁)) β†’ (((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) ∈ β„‚)
13896, 122, 123, 137fsumsplit 15684 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (1...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = (Σ𝑖 ∈ (1...2)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) + Σ𝑖 ∈ (3...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2)))
13995, 134, 1383eqtr4d 2783 . 2 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ Σ𝑖 ∈ (1...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (1...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2))
14065adantr 482 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
141128adantr 482 . . 3 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
142 brcgr 28148 . . 3 (((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) ∧ (𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘))) β†’ (βŸ¨π‘ƒ, 𝐴⟩CgrβŸ¨π‘„, 𝐴⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (1...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2)))
143140, 141, 87, 141, 142syl22anc 838 . 2 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ (βŸ¨π‘ƒ, 𝐴⟩CgrβŸ¨π‘„, 𝐴⟩ ↔ Σ𝑖 ∈ (1...𝑁)(((π‘ƒβ€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2) = Σ𝑖 ∈ (1...𝑁)(((π‘„β€˜π‘–) βˆ’ (π΄β€˜π‘–))↑2)))
144139, 143mpbird 257 1 ((𝑁 ∈ (β„€β‰₯β€˜3) ∧ 𝐼 ∈ (2...(𝑁 βˆ’ 1))) β†’ βŸ¨π‘ƒ, 𝐴⟩CgrβŸ¨π‘„, 𝐴⟩)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {csn 4628  {cpr 4630  βŸ¨cop 4634   class class class wbr 5148   Γ— cxp 5674   Fn wfn 6536  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   + caddc 11110   ≀ cle 11246   βˆ’ cmin 11441  -cneg 11442  β„•cn 12209  2c2 12264  3c3 12265  β„€cz 12555  β„€β‰₯cuz 12819  ...cfz 13481  β†‘cexp 14024  Ξ£csu 15629  π”Όcee 28136  Cgrccgr 28138
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-oi 9502  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-n0 12470  df-z 12556  df-uz 12820  df-rp 12972  df-fz 13482  df-fzo 13625  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-sum 15630  df-ee 28139  df-cgr 28141
This theorem is referenced by:  axlowdim  28209
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