Theorem iccpartrn 44851

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	⊢ (𝜑 → 𝑀 ∈ ℕ)
iccpartgtprec.p	⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))

Assertion

Ref	Expression
iccpartrn	⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))

Proof of Theorem iccpartrn

Dummy variables 𝑖 𝑘 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.p	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
2		iccpartgtprec.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		iccpart 44837	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
5		elmapfn 8636	. . . . . . 7 ⊢ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) → 𝑃 Fn (0...𝑀))
6	5	adantr 481	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
7	4, 6	syl6bi 252	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
8	1, 7	mpd 15	. . . 4 ⊢ (𝜑 → 𝑃 Fn (0...𝑀))
9		fvelrnb 6827	. . . 4 ⊢ (𝑃 Fn (0...𝑀) → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝))
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝))
11	2	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
12	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
14	11, 12, 13	iccpartxr 44840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ℝ*)
15	2, 1	iccpartgel 44850	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘))
16		fveq2 6771	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
17	16	breq2d 5091	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃‘𝑘) ↔ (𝑃‘0) ≤ (𝑃‘𝑖)))
18	17	rspcva 3559	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘)) → (𝑃‘0) ≤ (𝑃‘𝑖))
19	18	expcom 414	. . . . . . . 8 ⊢ (∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘) → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃‘𝑖)))
20	15, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃‘𝑖)))
21	20	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃‘𝑖))
22	2, 1	iccpartleu 44849	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀))
23	16	breq1d 5089	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) ≤ (𝑃‘𝑀) ↔ (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
24	23	rspcva 3559	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
25	24	expcom 414	. . . . . . . 8 ⊢ (∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
26	22, 25	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
27	26	imp 407	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
28		nnnn0 12240	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29		0elfz 13352	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
30	2, 28, 29	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑀))
31	2, 1, 30	iccpartxr 44840	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ*)
32		nn0fz0 13353	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (0...𝑀))
33	28, 32	sylib 217	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
34	2, 33	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
35	2, 1, 34	iccpartxr 44840	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*)
36	31, 35	jca 512	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
37	36	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
38		elicc1 13122	. . . . . . 7 ⊢ (((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*) → ((𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)) ↔ ((𝑃‘𝑖) ∈ ℝ* ∧ (𝑃‘0) ≤ (𝑃‘𝑖) ∧ (𝑃‘𝑖) ≤ (𝑃‘𝑀))))
39	37, 38	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)) ↔ ((𝑃‘𝑖) ∈ ℝ* ∧ (𝑃‘0) ≤ (𝑃‘𝑖) ∧ (𝑃‘𝑖) ≤ (𝑃‘𝑀))))
40	14, 21, 27, 39	mpbir3and 1341	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)))
41		eleq1 2828	. . . . 5 ⊢ ((𝑃‘𝑖) = 𝑝 → ((𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
42	40, 41	syl5ibcom 244	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘𝑖) = 𝑝 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
43	42	rexlimdva 3215	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
44	10, 43	sylbid 239	. 2 ⊢ (𝜑 → (𝑝 ∈ ran 𝑃 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
45	44	ssrdv 3932	1 ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067   ⊆ wss 3892   class class class wbr 5079  ran crn 5591   Fn wfn 6427  ‘cfv 6432  (class class class)co 7271   ↑_m cmap 8598  0cc0 10872  1c1 10873   + caddc 10875  ℝ^*cxr 11009   < clt 11010   ≤ cle 11011  ℕcn 11973  ℕ₀cn0 12233  [,]cicc 13081  ...cfz 13238  ..^cfzo 13381  RePartciccp 44834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-n0 12234  df-z 12320  df-uz 12582  df-icc 13085  df-fz 13239  df-fzo 13382  df-iccp 44835

This theorem is referenced by:  iccpartf  44852