Theorem iccpartrn 48207

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 48193	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
4	2, 3	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) ↔ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
5		elmapfn 8858	. . . . . . 7 ⊢ (𝑃 ∈ (ℝ* ↑m (0...𝑀)) → 𝑃 Fn (0...𝑀))
6	5	adantr 485	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 Fn (0...𝑀))
7	4, 6	biimtrdi 256	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ (RePart‘𝑀) → 𝑃 Fn (0...𝑀)))
8	1, 7	mpd 16	. . . 4 ⊢ (𝜑 → 𝑃 Fn (0...𝑀))
9		fvelrnb 6941	. . . 4 ⊢ (𝑃 Fn (0...𝑀) → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝))
10	8, 9	syl 18	. . 3 ⊢ (𝜑 → (𝑝 ∈ ran 𝑃 ↔ ∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝))
11	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑀 ∈ ℕ)
12	1	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
13		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → 𝑖 ∈ (0...𝑀))
14	11, 12, 13	iccpartxr 48196	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ℝ*)
15	2, 1	iccpartgel 48206	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘))
16		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
17	16	breq2d 5121	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘0) ≤ (𝑃‘𝑘) ↔ (𝑃‘0) ≤ (𝑃‘𝑖)))
18	17	rspcva 3579	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘)) → (𝑃‘0) ≤ (𝑃‘𝑖))
19	18	expcom 418	. . . . . . . 8 ⊢ (∀𝑘 ∈ (0...𝑀)(𝑃‘0) ≤ (𝑃‘𝑘) → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃‘𝑖)))
20	15, 19	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘0) ≤ (𝑃‘𝑖)))
21	20	imp 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘0) ≤ (𝑃‘𝑖))
22	2, 1	iccpartleu 48205	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀))
23	16	breq1d 5119	. . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) ≤ (𝑃‘𝑀) ↔ (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
24	23	rspcva 3579	. . . . . . . . 9 ⊢ ((𝑖 ∈ (0...𝑀) ∧ ∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
25	24	expcom 418	. . . . . . . 8 ⊢ (∀𝑘 ∈ (0...𝑀)(𝑃‘𝑘) ≤ (𝑃‘𝑀) → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
26	22, 25	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) ≤ (𝑃‘𝑀)))
27	26	imp 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ≤ (𝑃‘𝑀))
28		nnnn0 12515	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
29		0elfz 13657	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ0 → 0 ∈ (0...𝑀))
30	2, 28, 29	3syl 19	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑀))
31	2, 1, 30	iccpartxr 48196	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ*)
32		nn0fz0 13658	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (0...𝑀))
33	28, 32	sylib 221	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
34	2, 33	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (0...𝑀))
35	2, 1, 34	iccpartxr 48196	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘𝑀) ∈ ℝ*)
36	31, 35	jca 520	. . . . . . . 8 ⊢ (𝜑 → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
37	36	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘0) ∈ ℝ* ∧ (𝑃‘𝑀) ∈ ℝ*))
38		elicc1 13420	. . . . . . 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 18	. . . . . 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 1361	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → (𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)))
41		eleq1 2851	. . . . 5 ⊢ ((𝑃‘𝑖) = 𝑝 → ((𝑃‘𝑖) ∈ ((𝑃‘0)[,](𝑃‘𝑀)) ↔ 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
42	40, 41	syl5ibcom 248	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝑀)) → ((𝑃‘𝑖) = 𝑝 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
43	42	rexlimdva 3166	. . 3 ⊢ (𝜑 → (∃𝑖 ∈ (0...𝑀)(𝑃‘𝑖) = 𝑝 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
44	10, 43	sylbid 243	. 2 ⊢ (𝜑 → (𝑝 ∈ ran 𝑃 → 𝑝 ∈ ((𝑃‘0)[,](𝑃‘𝑀))))
45	44	ssrdv 3943	1 ⊢ (𝜑 → ran 𝑃 ⊆ ((𝑃‘0)[,](𝑃‘𝑀)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905   class class class wbr 5109  ran crn 5662   Fn wfn 6531  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8820  0cc0 11104  1c1 11105   + caddc 11107  ℝ^*cxr 11246   < clt 11247   ≤ cle 11248  ℕcn 12237  ℕ₀cn0 12508  [,]cicc 13379  ...cfz 13539  ..^cfzo 13687  RePartciccp 48190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  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 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-n0 12509  df-z 12596  df-uz 12867  df-icc 13383  df-fz 13540  df-fzo 13688  df-iccp 48191

This theorem is used by:  iccpartf  48208