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Theorem iccpartrn 46693
Description: If there is a partition, then all intermediate points and bounds are contained in a 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 46679 . . . . . . 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 8875 . . . . . . 7 (𝑃 ∈ (ℝ*m (0...𝑀)) → 𝑃 Fn (0...𝑀))
65adantr 480 . . . . . 6 ((𝑃 ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
74, 6syl6bi 253 . . . . 5 (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
81, 7mpd 15 . . . 4 (𝜑𝑃 Fn (0...𝑀))
9 fvelrnb 6953 . . . 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 480 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
121adantr 480 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13 simpr 484 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
1411, 12, 13iccpartxr 46682 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ℝ*)
152, 1iccpartgel 46692 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘))
16 fveq2 6891 . . . . . . . . . . 11 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
1716breq2d 5154 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃𝑘) ↔ (𝑃‘0) ≤ (𝑃𝑖)))
1817rspcva 3605 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃𝑘)) → (𝑃‘0) ≤ (𝑃𝑖))
1918expcom 413 . . . . . . . 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 406 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃𝑖))
222, 1iccpartleu 46691 . . . . . . . 8 (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀))
2316breq1d 5152 . . . . . . . . . 10 (𝑘 = 𝑖 → ((𝑃𝑘) ≤ (𝑃𝑀) ↔ (𝑃𝑖) ≤ (𝑃𝑀)))
2423rspcva 3605 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
2524expcom 413 . . . . . . . 8 (∀𝑘 ∈ (0...𝑀)(𝑃𝑘) ≤ (𝑃𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2622, 25syl 17 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃𝑖) ≤ (𝑃𝑀)))
2726imp 406 . . . . . 6 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ≤ (𝑃𝑀))
28 nnnn0 12501 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29 0elfz 13622 . . . . . . . . . . 11 (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
302, 28, 293syl 18 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑀))
312, 1, 30iccpartxr 46682 . . . . . . . . 9 (𝜑 → (𝑃‘0) ∈ ℝ*)
32 nn0fz0 13623 . . . . . . . . . . . 12 (𝑀 ∈ ℕ0𝑀 ∈ (0...𝑀))
3328, 32sylib 217 . . . . . . . . . . 11 (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
342, 33syl 17 . . . . . . . . . 10 (𝜑𝑀 ∈ (0...𝑀))
352, 1, 34iccpartxr 46682 . . . . . . . . 9 (𝜑 → (𝑃𝑀) ∈ ℝ*)
3631, 35jca 511 . . . . . . . 8 (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
3736adantr 480 . . . . . . 7 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃𝑀) ∈ ℝ*))
38 elicc1 13392 . . . . . . 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 1340 . . . . 5 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)))
41 eleq1 2816 . . . . 5 ((𝑃𝑖) = 𝑝 → ((𝑃𝑖) ∈ ((𝑃‘0)[,](𝑃𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4240, 41syl5ibcom 244 . . . 4 ((𝜑𝑖 ∈ (0...𝑀)) → ((𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4342rexlimdva 3150 . . 3 (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃𝑖) = 𝑝𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4410, 43sylbid 239 . 2 (𝜑 → (𝑝 ∈ ran 𝑃𝑝 ∈ ((𝑃‘0)[,](𝑃𝑀))))
4544ssrdv 3984 1 (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃𝑀)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  wrex 3065  wss 3944   class class class wbr 5142  ran crn 5673   Fn wfn 6537  cfv 6542  (class class class)co 7414  m cmap 8836  0cc0 11130  1c1 11131   + caddc 11133  *cxr 11269   < clt 11270  cle 11271  cn 12234  0cn0 12494  [,]cicc 13351  ...cfz 13508  ..^cfzo 13651  RePartciccp 46676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-z 12581  df-uz 12845  df-icc 13355  df-fz 13509  df-fzo 13652  df-iccp 46677
This theorem is referenced by:  iccpartf  46694
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