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Mirrors > Home > MPE Home > Th. List > pfxccatin12lem4 | Structured version Visualization version GIF version |
Description: Lemma 4 for pfxccatin12 14665. (Contributed by Alexander van der Vekens, 30-Mar-2018.) (Revised by Alexander van der Vekens, 23-May-2018.) |
Ref | Expression |
---|---|
pfxccatin12lem4 | ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → ((𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀))) → 𝐾 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 12565 | . . . . . 6 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℤ) | |
2 | nn0z 12565 | . . . . . 6 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℤ) | |
3 | zsubcl 12586 | . . . . . 6 ⊢ ((𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐿 − 𝑀) ∈ ℤ) | |
4 | 1, 2, 3 | syl2an 596 | . . . . 5 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0) → (𝐿 − 𝑀) ∈ ℤ) |
5 | 4 | 3adant3 1132 | . . . 4 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐿 − 𝑀) ∈ ℤ) |
6 | elfzonelfzo 13716 | . . . . 5 ⊢ ((𝐿 − 𝑀) ∈ ℤ → ((𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀))) → 𝐾 ∈ ((𝐿 − 𝑀)..^(𝑁 − 𝑀)))) | |
7 | 6 | imp 407 | . . . 4 ⊢ (((𝐿 − 𝑀) ∈ ℤ ∧ (𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀)))) → 𝐾 ∈ ((𝐿 − 𝑀)..^(𝑁 − 𝑀))) |
8 | 5, 7 | sylan 580 | . . 3 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀)))) → 𝐾 ∈ ((𝐿 − 𝑀)..^(𝑁 − 𝑀))) |
9 | nn0cn 12464 | . . . . . . 7 ⊢ (𝐿 ∈ ℕ0 → 𝐿 ∈ ℂ) | |
10 | nn0cn 12464 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → 𝑀 ∈ ℂ) | |
11 | zcn 12545 | . . . . . . 7 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
12 | npncan3 11480 | . . . . . . 7 ⊢ ((𝐿 ∈ ℂ ∧ 𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐿 − 𝑀) + (𝑁 − 𝐿)) = (𝑁 − 𝑀)) | |
13 | 9, 10, 11, 12 | syl3an 1160 | . . . . . 6 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → ((𝐿 − 𝑀) + (𝑁 − 𝐿)) = (𝑁 − 𝑀)) |
14 | 13 | oveq2d 7409 | . . . . 5 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))) = ((𝐿 − 𝑀)..^(𝑁 − 𝑀))) |
15 | 14 | eleq2d 2818 | . . . 4 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))) ↔ 𝐾 ∈ ((𝐿 − 𝑀)..^(𝑁 − 𝑀)))) |
16 | 15 | adantr 481 | . . 3 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀)))) → (𝐾 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))) ↔ 𝐾 ∈ ((𝐿 − 𝑀)..^(𝑁 − 𝑀)))) |
17 | 8, 16 | mpbird 256 | . 2 ⊢ (((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) ∧ (𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀)))) → 𝐾 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿)))) |
18 | 17 | ex 413 | 1 ⊢ ((𝐿 ∈ ℕ0 ∧ 𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → ((𝐾 ∈ (0..^(𝑁 − 𝑀)) ∧ ¬ 𝐾 ∈ (0..^(𝐿 − 𝑀))) → 𝐾 ∈ ((𝐿 − 𝑀)..^((𝐿 − 𝑀) + (𝑁 − 𝐿))))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 (class class class)co 7393 ℂcc 11090 0cc0 11092 + caddc 11095 − cmin 11426 ℕ0cn0 12454 ℤcz 12540 ..^cfzo 13609 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-fzo 13610 |
This theorem is referenced by: pfxccatin12 14665 |
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