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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 12625 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ) | |
| 3 | nnz 12634 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
| 4 | nngt0 12297 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀) | |
| 5 | 2, 3, 4 | 3jca 1129 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
| 6 | 1, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
| 7 | fzopred 47334 | . . . . . 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 12388 | . . . . . . . 8 ⊢ (0 + 1) = 1 | |
| 10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (0 + 1) = 1) |
| 11 | 10 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → ((0 + 1)..^𝑀) = (1..^𝑀)) |
| 12 | 11 | uneq2d 4168 | . . . . 5 ⊢ (𝜑 → ({0} ∪ ((0 + 1)..^𝑀)) = ({0} ∪ (1..^𝑀))) |
| 13 | 8, 12 | eqtrd 2777 | . . . 4 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ (1..^𝑀))) |
| 14 | 13 | eleq2d 2827 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ ({0} ∪ (1..^𝑀)))) |
| 15 | elun 4153 | . . . 4 ⊢ (𝑖 ∈ ({0} ∪ (1..^𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀))) | |
| 16 | elsni 4643 | . . . . . . 7 ⊢ (𝑖 ∈ {0} → 𝑖 = 0) | |
| 17 | fveq2 6906 | . . . . . . . . . 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 47411 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) |
| 21 | 20 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀)) |
| 22 | 18, 21 | eqbrtrd 5165 | . . . . . . . 8 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑀)) |
| 23 | 22 | ex 412 | . . . . . . 7 ⊢ (𝑖 = 0 → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
| 24 | 16, 23 | syl 17 | . . . . . 6 ⊢ (𝑖 ∈ {0} → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑀))) |
| 25 | fveq2 6906 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
| 26 | 25 | breq1d 5153 | . . . . . . . 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 47409 | . . . . . . 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 858 | . . . . 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 | biimtrid 242 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ({0} ∪ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
| 33 | 14, 32 | sylbid 240 | . 2 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))) |
| 34 | 33 | ralrimiv 3145 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∪ cun 3949 {csn 4626 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 < clt 11295 ℕcn 12266 ℤcz 12613 ..^cfzo 13694 RePartciccp 47400 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-iccp 47401 |
| This theorem is referenced by: iccpartleu 47415 |
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