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Theorem iccpartrn 40661
Description: If there is a partition, then all intermediate points and bounds are contained in an closed interval of extended reals. (Contributed by AV, 14-Jul-2020.)
Hypotheses
Ref Expression
iccpartgtprec.m (𝜑𝑀 ∈ ℕ)
iccpartgtprec.p (𝜑𝑃 ∈ (RePart‘𝑀))
Assertion
Ref Expression
iccpartrn (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))

Proof of Theorem iccpartrn
Dummy variables 𝑖 𝑘 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iccpartgtprec.p . . . . 5 (𝜑𝑃 ∈ (RePart‘𝑀))
2 iccpartgtprec.m . . . . . . 7 (𝜑𝑀 ∈ ℕ)
3 iccpart 40647 . . . . . . 7 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
42, 3syl 17 . . . . . 6 (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
5 elmapfn 7824 . . . . . . 7 (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) → 𝑃 Fn (0...𝑀))
65adantr 481 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
74, 6syl6bi 243 . . . . 5 (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
81, 7mpd 15 . . . 4 (𝜑𝑃 Fn (0...𝑀))
9 fvelrnb 6200 . . . 4 (𝑃 Fn (0...𝑀) → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝))
108, 9syl 17 . . 3 (𝜑 → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝))
112adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
121adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13 simpr 477 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
1411, 12, 13iccpartxr 40650 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ℝ*)
152, 1iccpartgel 40660 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘))
16 fveq2 6148 . . . . . . . . . . 11 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
1716breq2d 4625 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃𝑘) ↔ (𝑃‘0) ≤ (𝑃𝑖)))
1817rspcva 3293 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘)) → (𝑃‘0) ≤ (𝑃𝑖))
1918expcom 451 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘) → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃𝑖)))
2015, 19syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃𝑖)))
2120imp 445 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃𝑖))
222, 1iccpartleu 40659 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀))
2316breq1d 4623 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃𝑘) ≤ (𝑃𝑀) ↔ (𝑃𝑖) ≤ (𝑃𝑀)))
2423rspcva 3293 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
2524expcom 451 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2622, 25syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2726imp 445 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
28 nnnn0 11243 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29 0elfz 12377 . . . . . . . . . . 11 (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
302, 28, 293syl 18 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑀))
312, 1, 30iccpartxr 40650 . . . . . . . . 9 (𝜑 → (𝑃‘0) ∈ ℝ*)
32 nn0fz0 12378 . . . . . . . . . . . 12 (𝑀 ∈ ℕ0𝑀 ∈ (0...𝑀))
3328, 32sylib 208 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
342, 33syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (0...𝑀))
352, 1, 34iccpartxr 40650 . . . . . . . . 9 (𝜑 → (𝑃𝑀) ∈ ℝ*)
3631, 35jca 554 . . . . . . . 8 (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
3736adantr 481 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
38 elicc1 12161 . . . . . . 7 (((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*) → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘0) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ (𝑃𝑀))))
3937, 38syl 17 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘0) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ (𝑃𝑀))))
4014, 21, 27, 39mpbir3and 1243 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)))
41 eleq1 2686 . . . . 5 ((𝑃𝑖) = 𝑝 → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4240, 41syl5ibcom 235 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4342rexlimdva 3024 . . 3 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4410, 43sylbid 230 . 2 (𝜑 → (𝑝 ∈ ran 𝑃𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4544ssrdv 3589 1 (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555   class class class wbr 4613  ran crn 5075   Fn wfn 5842  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  0cc0 9880  1c1 9881   + caddc 9883  *cxr 10017   < clt 10018  cle 10019  cn 10964  0cn0 11236  [,]cicc 12120  ...cfz 12268  ..^cfzo 12406  RePartciccp 40644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-icc 12124  df-fz 12269  df-fzo 12407  df-iccp 40645
This theorem is referenced by:  iccpartf  40662
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