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Theorem iccpartres 41781
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 11145 . . . 4 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2 iccpart 41779 . . . 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 474 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))))
5 nnz 11512 . . . . . . . . 9 (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6 uzid 11815 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
75, 6syl 17 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ𝑀))
8 peano2uz 11855 . . . . . . . 8 (𝑀 ∈ (ℤ𝑀) → (𝑀 + 1) ∈ (ℤ𝑀))
97, 8syl 17 . . . . . . 7 (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ𝑀))
10 fzss2 12495 . . . . . . 7 ((𝑀 + 1) ∈ (ℤ𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
119, 10syl 17 . . . . . 6 (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12 elmapssres 7999 . . . . . 6 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ*𝑚 (0...𝑀)))
134, 11, 12syl2anr 496 . . . . 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 12611 . . . . . . . . . 10 ((𝑀 + 1) ∈ (ℤ𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
159, 14syl 17 . . . . . . . . 9 (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16 ssralv 3772 . . . . . . . . 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 484 . . . . . . 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 444 . . . . . 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 12603 . . . . . . . . . . . . . . 15 (0..^𝑀) ⊆ (0...𝑀)
2120a1i 11 . . . . . . . . . . . . . 14 ((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (0..^𝑀) ⊆ (0...𝑀))
2221sselda 3709 . . . . . . . . . . . . 13 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23 fvres 6320 . . . . . . . . . . . . . 14 (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃𝑖))
2423eqcomd 2730 . . . . . . . . . . . . 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 12589 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ‘0))
2827adantl 473 . . . . . . . . . . . . . . . 16 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ‘0))
29 fzofzp1b 12681 . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . 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 2730 . . . . . . . . . . . 12 (((𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
3525, 34breq12d 4773 . . . . . . . . . . 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 3064 . . . . . . . . 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 449 . . . . . . . 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 472 . . . . . . 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 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)) → ∀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 41779 . . . . . 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 472 . . . . 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 995 . . . 4 ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ*𝑚 (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
4544ex 449 . . 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 444 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 383   = wceq 1596  wcel 2103  wral 3014  wss 3680   class class class wbr 4760  cres 5220  cfv 6001  (class class class)co 6765  𝑚 cmap 7974  0cc0 10049  1c1 10050   + caddc 10052  *cxr 10186   < clt 10187  cn 11133  cz 11490  cuz 11800  ...cfz 12440  ..^cfzo 12580  RePartciccp 41776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-iccp 41777
This theorem is referenced by:  iccelpart  41796
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