Theorem iccpartrn 46396

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 46382	. . . . . . 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 8861	. . . . . . 7 ⊢ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) → 𝑃 Fn (0...𝑀))
6	5	adantr 479	. . . . . 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 6951	. . . 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 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
12	1	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
14	11, 12, 13	iccpartxr 46385	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ℝ*)
15	2, 1	iccpartgel 46395	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘))
16		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
17	16	breq2d 5159	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃‘𝑘) ↔ (𝑃‘0) ≤ (𝑃‘𝑖)))
18	17	rspcva 3609	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘)) → (𝑃‘0) ≤ (𝑃‘𝑖))
19	18	expcom 412	. . . . . . . 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 405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃‘𝑖))
22	2, 1	iccpartleu 46394	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀))
23	16	breq1d 5157	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) ≤ (𝑃‘𝑀) ↔ (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
24	23	rspcva 3609	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
25	24	expcom 412	. . . . . . . 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 405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
28		nnnn0 12483	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29		0elfz 13602	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
30	2, 28, 29	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑀))
31	2, 1, 30	iccpartxr 46385	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ*)
32		nn0fz0 13603	. . . . . . . . . . . 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 46385	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*)
36	31, 35	jca 510	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
37	36	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
38		elicc1 13372	. . . . . . 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 1340	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)))
41		eleq1 2819	. . . . 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 3153	. . 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 3987	1 ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   ⊆ wss 3947   class class class wbr 5147  ran crn 5676   Fn wfn 6537  ‘cfv 6542  (class class class)co 7411   ↑_m cmap 8822  0cc0 11112  1c1 11113   + caddc 11115  ℝ^*cxr 11251   < clt 11252   ≤ cle 11253  ℕcn 12216  ℕ₀cn0 12476  [,]cicc 13331  ...cfz 13488  ..^cfzo 13631  RePartciccp 46379

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-icc 13335  df-fz 13489  df-fzo 13632  df-iccp 46380

This theorem is referenced by:  iccpartf  46397