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Theorem iccpartres 40652
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 10976 . . . 4 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2 iccpart 40650 . . . 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 473 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))))
5 nnz 11343 . . . . . . . . 9 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6 uzid 11646 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
75, 6syl 17 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ𝑀))
8 peano2uz 11685 . . . . . . . 8 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
97, 8syl 17 . . . . . . 7 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ𝑀))
10 fzss2 12323 . . . . . . 7 ((𝑀 + 1) ∈ (ℤ𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
119, 10syl 17 . . . . . 6 (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12 elmapssres 7826 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ*𝑚 (0...𝑀)))
134, 11, 12syl2anr 495 . . . . 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 12437 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
159, 14syl 17 . . . . . . . . 9 (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16 ssralv 3645 . . . . . . . . 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 483 . . . . . . 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 445 . . . . . 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 12429 . . . . . . . . . . . . . . 15 (0..^𝑀) ⊆ (0...𝑀)
2120a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (0..^𝑀) ⊆ (0...𝑀))
2221sselda 3583 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23 fvres 6164 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃𝑖))
2423eqcomd 2627 . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . 15 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27 elfzouz 12415 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ‘0))
2827adantl 482 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ‘0))
29 fzofzp1b 12507 . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 14 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32 fvres 6164 . . . . . . . . . . . . . 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 2627 . . . . . . . . . . . 12 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
3525, 34breq12d 4626 . . . . . . . . . . 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 219 . . . . . . . . . 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 2956 . . . . . . . . 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 450 . . . . . . . 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 481 . . . . . . 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 446 . . . . . 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 40650 . . . . . 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 481 . . . . 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 956 . . . 4 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
4544ex 450 . . 3 (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
463, 45sylbid 230 . 2 (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
4746imp 445 1 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3555   class class class wbr 4613  cres 5076  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  0cc0 9880  1c1 9881   + caddc 9883  *cxr 10017   < clt 10018  cn 10964  cz 11321  cuz 11631  ...cfz 12268  ..^cfzo 12406  RePartciccp 40647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-iccp 40648
This theorem is referenced by:  iccelpart  40667
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