Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartltu | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and the lower bound are strictly less than the upper bound. (Contributed by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
iccpartgtprec.p | ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) |
Ref | Expression |
---|---|
iccpartltu | ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
2 | 0zd 11266 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ) | |
3 | nnz 11276 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | nngt0 10926 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀) | |
5 | 2, 3, 4 | 3jca 1235 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
6 | 1, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
7 | fzopred 39945 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀) → (0..^𝑀) = ({0} ∪ ((0 + 1)..^𝑀))) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ ((0 + 1)..^𝑀))) |
9 | 0p1e1 11009 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (0 + 1) = 1) |
11 | 10 | oveq1d 6564 | . . . . . 6 ⊢ (𝜑 → ((0 + 1)..^𝑀) = (1..^𝑀)) |
12 | 11 | uneq2d 3729 | . . . . 5 ⊢ (𝜑 → ({0} ∪ ((0 + 1)..^𝑀)) = ({0} ∪ (1..^𝑀))) |
13 | 8, 12 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ (1..^𝑀))) |
14 | 13 | eleq2d 2673 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ ({0} ∪ (1..^𝑀)))) |
15 | elun 3715 | . . . 4 ⊢ (𝑖 ∈ ({0} ∪ (1..^𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀))) | |
16 | elsni 4142 | . . . . . . 7 ⊢ (𝑖 ∈ {0} → 𝑖 = 0) | |
17 | fveq2 6103 | . . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0)) | |
18 | 17 | adantr 480 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘0)) |
19 | iccpartgtprec.p | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
20 | 1, 19 | iccpartlt 39962 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) |
21 | 20 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀)) |
22 | 18, 21 | eqbrtrd 4605 | . . . . . . . 8 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
23 | 22 | ex 449 | . . . . . . 7 ⊢ (𝑖 = 0 → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
24 | 16, 23 | syl 17 | . . . . . 6 ⊢ (𝑖 ∈ {0} → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
25 | 1, 19 | iccpartiltu 39960 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀)) |
26 | fveq2 6103 | . . . . . . . . . 10 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
27 | 26 | breq1d 4593 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀))) |
28 | 27 | rspccv 3279 | . . . . . . . 8 ⊢ (∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀) → (𝑖 ∈ (1..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
29 | 25, 28 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑖 ∈ (1..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
30 | 29 | com12 32 | . . . . . 6 ⊢ (𝑖 ∈ (1..^𝑀) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
31 | 24, 30 | jaoi 393 | . . . . 5 ⊢ ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀)) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
32 | 31 | com12 32 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
33 | 15, 32 | syl5bi 231 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ({0} ∪ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
34 | 14, 33 | sylbid 229 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
35 | 34 | ralrimiv 2948 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ cun 3538 {csn 4125 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 < clt 9953 ℕcn 10897 ℤcz 11254 ..^cfzo 12334 RePartciccp 39951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-iccp 39952 |
This theorem is referenced by: iccpartleu 39966 |
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