Theorem iccpartres 47878

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.

Step	Hyp	Ref	Expression
1		peano2nn 12186	. . . 4 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
2		iccpart 47876	. . . 4 ⊢ ((𝑀 + 1) ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
3	1, 2	syl 17	. . 3 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) ↔ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))))
4		simpl 482	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → 𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))))
5		nnz 12545	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ)
6		uzid 12803	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
7	5, 6	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (ℤ≥‘𝑀))
8		peano2uz 12851	. . . . . . . 8 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘𝑀))
10		fzss2 13518	. . . . . . 7 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (0...𝑀) ⊆ (0...(𝑀 + 1)))
11	9, 10	syl 17	. . . . . 6 ⊢ (𝑀 ∈ ℕ → (0...𝑀) ⊆ (0...(𝑀 + 1)))
12		elmapssres 8814	. . . . . 6 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ (0...𝑀) ⊆ (0...(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑m (0...𝑀)))
13	4, 11, 12	syl2anr 598	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (ℝ* ↑m (0...𝑀)))
14		fzoss2 13642	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ≥‘𝑀) → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
15	9, 14	syl 17	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → (0..^𝑀) ⊆ (0..^(𝑀 + 1)))
16		ssralv 3990	. . . . . . . . 9 ⊢ ((0..^𝑀) ⊆ (0..^(𝑀 + 1)) → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
17	15, 16	syl 17	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
18	17	adantld 490	. . . . . . 7 ⊢ (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))))
19	18	imp 406	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))
20		fzossfz 13633	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑀) ⊆ (0...𝑀)
21	20	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (0..^𝑀) ⊆ (0...𝑀))
22	21	sselda 3921	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
23		fvres 6859	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘𝑖) = (𝑃‘𝑖))
24	23	eqcomd 2742	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ (0...𝑀) → (𝑃‘𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
25	22, 24	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) = ((𝑃 ↾ (0...𝑀))‘𝑖))
26		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0..^𝑀))
27		elfzouz 13618	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (ℤ≥‘0))
28	27	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (ℤ≥‘0))
29		fzofzp1b 13720	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (ℤ≥‘0) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
30	28, 29	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 ∈ (0..^𝑀) ↔ (𝑖 + 1) ∈ (0...𝑀)))
31	26, 30	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
32		fvres 6859	. . . . . . . . . . . . . 14 ⊢ ((𝑖 + 1) ∈ (0...𝑀) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
33	31, 32	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
34	33	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) = ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
35	25, 34	breq12d 5098	. . . . . . . . . . 11 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) ↔ ((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
36	35	biimpd 229	. . . . . . . . . 10 ⊢ (((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
37	36	ralimdva 3149	. . . . . . . . 9 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ 𝑀 ∈ ℕ) → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
38	37	ex 412	. . . . . . . 8 ⊢ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) → (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
39	38	adantr 480	. . . . . . 7 ⊢ ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝑀 ∈ ℕ → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))))
40	39	impcom 407	. . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → (∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1))))
41	19, 40	mpd 15	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → ∀𝑖 ∈ (0..^𝑀)((𝑃 ↾ (0...𝑀))‘𝑖) < ((𝑃 ↾ (0...𝑀))‘(𝑖 + 1)))
42		iccpart 47876	. . . . . 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)))))
43	42	adantr 480	. . . . 5 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (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)))))
44	13, 41, 43	mpbir2and 714	. . . 4 ⊢ ((𝑀 ∈ ℕ ∧ (𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1)))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))
45	44	ex 412	. . 3 ⊢ (𝑀 ∈ ℕ → ((𝑃 ∈ (ℝ* ↑m (0...(𝑀 + 1))) ∧ ∀𝑖 ∈ (0..^(𝑀 + 1))(𝑃‘𝑖) < (𝑃‘(𝑖 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
46	3, 45	sylbid 240	. 2 ⊢ (𝑀 ∈ ℕ → (𝑃 ∈ (RePart‘(𝑀 + 1)) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀)))
47	46	imp 406	1 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘(𝑀 + 1))) → (𝑃 ↾ (0...𝑀)) ∈ (RePart‘𝑀))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889   class class class wbr 5085   ↾ cres 5633  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773  0cc0 11038  1c1 11039   + caddc 11041  ℝ^*cxr 11178   < clt 11179  ℕcn 12174  ℤcz 12524  ℤ_≥cuz 12788  ...cfz 13461  ..^cfzo 13608  RePartciccp 47873

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-n0 12438  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-iccp 47874

This theorem is referenced by:  iccelpart  47893