Theorem iccpartrn 47411

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 47397	. . . . . . 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 8884	. . . . . . 7 ⊢ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) → 𝑃 Fn (0...𝑀))
6	5	adantr 480	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
7	4, 6	biimtrdi 253	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
8	1, 7	mpd 15	. . . 4 ⊢ (𝜑 → 𝑃 Fn (0...𝑀))
9		fvelrnb 6944	. . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
12	1	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
14	11, 12, 13	iccpartxr 47400	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ℝ*)
15	2, 1	iccpartgel 47410	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘))
16		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
17	16	breq2d 5136	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃‘𝑘) ↔ (𝑃‘0) ≤ (𝑃‘𝑖)))
18	17	rspcva 3604	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘)) → (𝑃‘0) ≤ (𝑃‘𝑖))
19	18	expcom 413	. . . . . . . 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 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃‘𝑖))
22	2, 1	iccpartleu 47409	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀))
23	16	breq1d 5134	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) ≤ (𝑃‘𝑀) ↔ (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
24	23	rspcva 3604	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
25	24	expcom 413	. . . . . . . 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 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
28		nnnn0 12513	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29		0elfz 13646	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
30	2, 28, 29	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑀))
31	2, 1, 30	iccpartxr 47400	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ*)
32		nn0fz0 13647	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (0...𝑀))
33	28, 32	sylib 218	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
34	2, 33	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
35	2, 1, 34	iccpartxr 47400	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*)
36	31, 35	jca 511	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
37	36	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
38		elicc1 13411	. . . . . . 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 1343	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)))
41		eleq1 2823	. . . . 5 ⊢ ((𝑃‘𝑖) = 𝑝 → ((𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
42	40, 41	syl5ibcom 245	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘𝑖) = 𝑝 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
43	42	rexlimdva 3142	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
44	10, 43	sylbid 240	. 2 ⊢ (𝜑 → (𝑝 ∈ ran 𝑃 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
45	44	ssrdv 3969	1 ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061   ⊆ wss 3931   class class class wbr 5124  ran crn 5660   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845  0cc0 11134  1c1 11135   + caddc 11137  ℝ^*cxr 11273   < clt 11274   ≤ cle 11275  ℕcn 12245  ℕ₀cn0 12506  [,]cicc 13370  ...cfz 13529  ..^cfzo 13676  RePartciccp 47394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-n0 12507  df-z 12594  df-uz 12858  df-icc 13374  df-fz 13530  df-fzo 13677  df-iccp 47395

This theorem is referenced by:  iccpartf  47412