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Theorem iccpartres 43878
Description: The restriction of a partition is a partition. (Contributed by AV, 16-Jul-2020.)
Assertion
Ref Expression
iccpartres ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))

Proof of Theorem iccpartres
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 peano2nn 11637 . . . 4 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2 iccpart 43876 . . . 4 ((𝑀 + 1) ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
31, 2syl 17 . . 3 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))))
4 simpl 486 . . . . . 6 ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*m (0...(𝑀 + 1))))
5 nnz 11992 . . . . . . . . 9 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6 uzid 12246 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
75, 6syl 17 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ𝑀))
8 peano2uz 12289 . . . . . . . 8 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
97, 8syl 17 . . . . . . 7 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ𝑀))
10 fzss2 12942 . . . . . . 7 ((𝑀 + 1) ∈ (ℤ𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
119, 10syl 17 . . . . . 6 (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12 elmapssres 8418 . . . . . 6 ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ*m (0...𝑀)))
134, 11, 12syl2anr 599 . . . . 5 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ*m (0...𝑀)))
14 fzoss2 13060 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
159, 14syl 17 . . . . . . . . 9 (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16 ssralv 4008 . . . . . . . . 9 ((0..^𝑀) ⊆ (0..^(𝑀 + 1)) → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
1715, 16syl 17 . . . . . . . 8 (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
1817adantld 494 . . . . . . 7 (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1))))
1918imp 410 . . . . . 6 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → ∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)))
20 fzossfz 13051 . . . . . . . . . . . . . . 15 (0..^𝑀) ⊆ (0...𝑀)
2120a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (0..^𝑀) ⊆ (0...𝑀))
2221sselda 3942 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23 fvres 6671 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃𝑖))
2423eqcomd 2828 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) → (𝑃𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
2522, 24syl 17 . . . . . . . . . . . 12 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
26 simpr 488 . . . . . . . . . . . . . . 15 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27 elfzouz 13037 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ‘0))
2827adantl 485 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ‘0))
29 fzofzp1b 13130 . . . . . . . . . . . . . . . 16 (𝑖 ∈ (ℤ‘0) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
3028, 29syl 17 . . . . . . . . . . . . . . 15 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
3126, 30mpbid 235 . . . . . . . . . . . . . 14 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32 fvres 6671 . . . . . . . . . . . . . 14 ((𝑖 + 1) ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
3331, 32syl 17 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
3433eqcomd 2828 . . . . . . . . . . . 12 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
3525, 34breq12d 5055 . . . . . . . . . . 11 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃𝑖) < (𝑃‘(𝑖 + 1)) ↔ ((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
3635biimpd 232 . . . . . . . . . 10 (((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
3736ralimdva 3169 . . . . . . . . 9 ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
3837ex 416 . . . . . . . 8 (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) → (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
3938adantr 484 . . . . . . 7 ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
4039impcom 411 . . . . . 6 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → (∀𝑖 ∈ (0..^𝑀)(𝑃𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
4119, 40mpd 15 . . . . 5 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
42 iccpart 43876 . . . . . 6 (𝑀 ∈ ℕ → ((𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀) ↔ ((𝑃 ↾ (0...𝑀)) ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
4342adantr 484 . . . . 5 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → ((𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀) ↔ ((𝑃 ↾ (0...𝑀)) ∈ (ℝ*m (0...𝑀)) ∧ ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
4413, 41, 43mpbir2and 712 . . . 4 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
4544ex 416 . . 3 (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ*m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
463, 45sylbid 243 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
4746imp 410 1 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wral 3130  wss 3908   class class class wbr 5042  cres 5534  cfv 6334  (class class class)co 7140  m cmap 8393  0cc0 10526  1c1 10527   + caddc 10529  *cxr 10663   < clt 10664  cn 11625  cz 11969  cuz 12231  ...cfz 12885  ..^cfzo 13028  RePartciccp 43873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-iccp 43874
This theorem is referenced by:  iccelpart  43893
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