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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem1 | Structured version Visualization version GIF version |
Description: Lemma for submateq 32777. (Contributed by Thierry Arnoux, 25-Aug-2020.) |
Ref | Expression |
---|---|
submateqlem1.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
submateqlem1.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
submateqlem1.m | ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
submateqlem1.1 | ⊢ (𝜑 → 𝐾 ≤ 𝑀) |
Ref | Expression |
---|---|
submateqlem1 | ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1ssnn 13528 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
2 | submateqlem1.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
3 | 1, 2 | sselid 3979 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
4 | 3 | nnzd 12581 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | submateqlem1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnzd 12581 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | fz1ssnn 13528 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
8 | submateqlem1.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
9 | 7, 8 | sselid 3979 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
10 | 9 | nnzd 12581 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
11 | submateqlem1.1 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑀) | |
12 | 9 | nnred 12223 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
13 | 5 | nnred 12223 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
14 | 1red 11211 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
15 | 13, 14 | resubcld 11638 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
16 | elfzle2 13501 | . . . . 5 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) | |
17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) |
18 | 13 | lem1d 12143 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ≤ 𝑁) |
19 | 12, 15, 13, 17, 18 | letrd 11367 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
20 | 4, 6, 10, 11, 19 | elfzd 13488 | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾...𝑁)) |
21 | 1zzd 12589 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
22 | 10 | peano2zd 12665 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
23 | 9 | nnnn0d 12528 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
24 | 23 | nn0ge0d 12531 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀) |
25 | 1re 11210 | . . . . . 6 ⊢ 1 ∈ ℝ | |
26 | addge02 11721 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ≤ 𝑀 ↔ 1 ≤ (𝑀 + 1))) | |
27 | 25, 12, 26 | sylancr 587 | . . . . 5 ⊢ (𝜑 → (0 ≤ 𝑀 ↔ 1 ≤ (𝑀 + 1))) |
28 | 24, 27 | mpbid 231 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝑀 + 1)) |
29 | 5 | nnnn0d 12528 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
30 | nn0ltlem1 12618 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | |
31 | 23, 29, 30 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
32 | 17, 31 | mpbird 256 | . . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁) |
33 | nnltp1le 12614 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | |
34 | 9, 5, 33 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
35 | 32, 34 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≤ 𝑁) |
36 | 21, 6, 22, 28, 35 | elfzd 13488 | . . 3 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁)) |
37 | 3 | nnred 12223 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
38 | nnleltp1 12613 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐾 ≤ 𝑀 ↔ 𝐾 < (𝑀 + 1))) | |
39 | 3, 9, 38 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → (𝐾 ≤ 𝑀 ↔ 𝐾 < (𝑀 + 1))) |
40 | 11, 39 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → 𝐾 < (𝑀 + 1)) |
41 | 37, 40 | ltned 11346 | . . . . 5 ⊢ (𝜑 → 𝐾 ≠ (𝑀 + 1)) |
42 | 41 | necomd 2996 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≠ 𝐾) |
43 | nelsn 4667 | . . . 4 ⊢ ((𝑀 + 1) ≠ 𝐾 → ¬ (𝑀 + 1) ∈ {𝐾}) | |
44 | 42, 43 | syl 17 | . . 3 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ {𝐾}) |
45 | 36, 44 | eldifd 3958 | . 2 ⊢ (𝜑 → (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})) |
46 | 20, 45 | jca 512 | 1 ⊢ (𝜑 → (𝑀 ∈ (𝐾...𝑁) ∧ (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3944 {csn 4627 class class class wbr 5147 (class class class)co 7405 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 − cmin 11440 ℕcn 12208 ℕ0cn0 12468 ...cfz 13480 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 |
This theorem is referenced by: submateq 32777 |
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