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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 12623 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 ∈ ℤ) | |
3 | nnz 12632 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
4 | nngt0 12295 | . . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 0 < 𝑀) | |
5 | 2, 3, 4 | 3jca 1127 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
6 | 1, 5 | syl 17 | . . . . . 6 ⊢ (𝜑 → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀)) |
7 | fzopred 47272 | . . . . . 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 12386 | . . . . . . . 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 4178 | . . . . 5 ⊢ (𝜑 → ({0} ∪ ((0 + 1)..^𝑀)) = ({0} ∪ (1..^𝑀))) |
13 | 8, 12 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → (0..^𝑀) = ({0} ∪ (1..^𝑀))) |
14 | 13 | eleq2d 2825 | . . 3 ⊢ (𝜑 → (𝑖 ∈ (0..^𝑀) ↔ 𝑖 ∈ ({0} ∪ (1..^𝑀)))) |
15 | elun 4163 | . . . 4 ⊢ (𝑖 ∈ ({0} ∪ (1..^𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1..^𝑀))) | |
16 | elsni 4648 | . . . . . . 7 ⊢ (𝑖 ∈ {0} → 𝑖 = 0) | |
17 | fveq2 6907 | . . . . . . . . . 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 47349 | . . . . . . . . . 10 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀)) |
21 | 20 | adantl 481 | . . . . . . . . 9 ⊢ ((𝑖 = 0 ∧ 𝜑) → (𝑃‘0) < (𝑃‘𝑀)) |
22 | 18, 21 | eqbrtrd 5170 | . . . . . . . 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 6907 | . . . . . . . . 9 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖)) | |
26 | 25 | breq1d 5158 | . . . . . . . 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 47347 | . . . . . . 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 857 | . . . . 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 3143 | 1 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∪ cun 3961 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 + caddc 11156 < clt 11293 ℕcn 12264 ℤcz 12611 ..^cfzo 13691 RePartciccp 47338 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-iccp 47339 |
This theorem is referenced by: iccpartleu 47353 |
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