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Theorem iccpartres 42242
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 11364 . . . 4 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2 iccpart 42240 . . . 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 476 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))))
5 nnz 11727 . . . . . . . . 9 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6 uzid 11983 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
75, 6syl 17 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ𝑀))
8 peano2uz 12023 . . . . . . . 8 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
97, 8syl 17 . . . . . . 7 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ𝑀))
10 fzss2 12674 . . . . . . 7 ((𝑀 + 1) ∈ (ℤ𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
119, 10syl 17 . . . . . 6 (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12 elmapssres 8147 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ*𝑚 (0...𝑀)))
134, 11, 12syl2anr 592 . . . . 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 12791 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
159, 14syl 17 . . . . . . . . 9 (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16 ssralv 3891 . . . . . . . . 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 486 . . . . . . 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 397 . . . . . 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 12783 . . . . . . . . . . . . . . 15 (0..^𝑀) ⊆ (0...𝑀)
2120a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (0..^𝑀) ⊆ (0...𝑀))
2221sselda 3827 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23 fvres 6452 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃𝑖))
2423eqcomd 2831 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 15 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27 elfzouz 12769 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ‘0))
2827adantl 475 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ‘0))
29 fzofzp1b 12861 . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 14 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32 fvres 6452 . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . 12 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
3525, 34breq12d 4886 . . . . . . . . . . 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 221 . . . . . . . . . 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 3171 . . . . . . . . 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 403 . . . . . . . 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 474 . . . . . . 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 398 . . . . . 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 42240 . . . . . 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 474 . . . . 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 706 . . . 4 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
4544ex 403 . . 3 (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
463, 45sylbid 232 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
4746imp 397 1 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3117  wss 3798   class class class wbr 4873  cres 5344  cfv 6123  (class class class)co 6905  𝑚 cmap 8122  0cc0 10252  1c1 10253   + caddc 10255  *cxr 10390   < clt 10391  cn 11350  cz 11704  cuz 11968  ...cfz 12619  ..^cfzo 12760  RePartciccp 42237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-fzo 12761  df-iccp 42238
This theorem is referenced by:  iccelpart  42257
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