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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 11987 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ) | |
3 | nnz 11998 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | nngt0 11662 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀) | |
5 | 2, 3, 4 | 3jca 1124 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
6 | 1, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
7 | fzopred 43516 | . . . . . 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 11753 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (0 + 1) = 1) |
11 | 10 | oveq1d 7165 | . . . . . 6 ⊢ (𝜑 → ((0 + 1)..^𝑀) = (1..^𝑀)) |
12 | 11 | uneq2d 4138 | . . . . 5 ⊢ (𝜑 → ({0} ∪ ((0 + 1)..^𝑀)) = ({0} ∪ (1..^𝑀))) |
13 | 8, 12 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ (1..^𝑀))) |
14 | 13 | eleq2d 2898 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ ({0} ∪ (1..^𝑀)))) |
15 | elun 4124 | . . . 4 ⊢ (𝑖 ∈ ({0} ∪ (1..^𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀))) | |
16 | elsni 4577 | . . . . . . 7 ⊢ (𝑖 ∈ {0} → 𝑖 = 0) | |
17 | fveq2 6664 | . . . . . . . . . 10 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0)) | |
18 | 17 | adantr 483 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) = (𝑃‘0)) |
19 | iccpartgtprec.p | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀)) | |
20 | 1, 19 | iccpartlt 43578 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) |
21 | 20 | adantl 484 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀)) |
22 | 18, 21 | eqbrtrd 5080 | . . . . . . . 8 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
23 | 22 | ex 415 | . . . . . . 7 ⊢ (𝑖 = 0 → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
24 | 16, 23 | syl 17 | . . . . . 6 ⊢ (𝑖 ∈ {0} → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
25 | fveq2 6664 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
26 | 25 | breq1d 5068 | . . . . . . . 8 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀))) |
27 | 26 | rspccv 3619 | . . . . . . 7 ⊢ (∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀) → (𝑖 ∈ (1..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
28 | 1, 19 | iccpartiltu 43576 | . . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀)) |
29 | 27, 28 | syl11 33 | . . . . . 6 ⊢ (𝑖 ∈ (1..^𝑀) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
30 | 24, 29 | jaoi 853 | . . . . 5 ⊢ ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀)) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
31 | 30 | com12 32 | . . . 4 ⊢ (𝜑 → ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
32 | 15, 31 | syl5bi 244 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ({0} ∪ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
33 | 14, 32 | sylbid 242 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
34 | 33 | ralrimiv 3181 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∪ cun 3933 {csn 4560 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 ℕcn 11632 ℤcz 11975 ..^cfzo 13027 RePartciccp 43567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-iccp 43568 |
This theorem is referenced by: iccpartleu 43582 |
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