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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem1 | Structured version Visualization version GIF version |
Description: Lemma for submateq 33333. (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 13550 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
2 | submateqlem1.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
3 | 1, 2 | sselid 3976 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
4 | 3 | nnzd 12601 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | submateqlem1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnzd 12601 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | fz1ssnn 13550 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
8 | submateqlem1.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
9 | 7, 8 | sselid 3976 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
10 | 9 | nnzd 12601 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
11 | submateqlem1.1 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑀) | |
12 | 9 | nnred 12243 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
13 | 5 | nnred 12243 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
14 | 1red 11231 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
15 | 13, 14 | resubcld 11658 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
16 | elfzle2 13523 | . . . . 5 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) | |
17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) |
18 | 13 | lem1d 12163 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ≤ 𝑁) |
19 | 12, 15, 13, 17, 18 | letrd 11387 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
20 | 4, 6, 10, 11, 19 | elfzd 13510 | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾...𝑁)) |
21 | 1zzd 12609 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
22 | 10 | peano2zd 12685 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
23 | 9 | nnnn0d 12548 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
24 | 23 | nn0ge0d 12551 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀) |
25 | 1re 11230 | . . . . . 6 ⊢ 1 ∈ ℝ | |
26 | addge02 11741 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ≤ 𝑀 ↔ 1 ≤ (𝑀 + 1))) | |
27 | 25, 12, 26 | sylancr 586 | . . . . 5 ⊢ (𝜑 → (0 ≤ 𝑀 ↔ 1 ≤ (𝑀 + 1))) |
28 | 24, 27 | mpbid 231 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝑀 + 1)) |
29 | 5 | nnnn0d 12548 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
30 | nn0ltlem1 12638 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | |
31 | 23, 29, 30 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
32 | 17, 31 | mpbird 257 | . . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁) |
33 | nnltp1le 12634 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | |
34 | 9, 5, 33 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
35 | 32, 34 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≤ 𝑁) |
36 | 21, 6, 22, 28, 35 | elfzd 13510 | . . 3 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁)) |
37 | 3 | nnred 12243 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
38 | nnleltp1 12633 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐾 ≤ 𝑀 ↔ 𝐾 < (𝑀 + 1))) | |
39 | 3, 9, 38 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → (𝐾 ≤ 𝑀 ↔ 𝐾 < (𝑀 + 1))) |
40 | 11, 39 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → 𝐾 < (𝑀 + 1)) |
41 | 37, 40 | ltned 11366 | . . . . 5 ⊢ (𝜑 → 𝐾 ≠ (𝑀 + 1)) |
42 | 41 | necomd 2991 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≠ 𝐾) |
43 | nelsn 4664 | . . . 4 ⊢ ((𝑀 + 1) ≠ 𝐾 → ¬ (𝑀 + 1) ∈ {𝐾}) | |
44 | 42, 43 | syl 17 | . . 3 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ {𝐾}) |
45 | 36, 44 | eldifd 3955 | . 2 ⊢ (𝜑 → (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})) |
46 | 20, 45 | jca 511 | 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 395 ∈ wcel 2099 ≠ wne 2935 ∖ cdif 3941 {csn 4624 class class class wbr 5142 (class class class)co 7414 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 < clt 11264 ≤ cle 11265 − cmin 11460 ℕcn 12228 ℕ0cn0 12488 ...cfz 13502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 |
This theorem is referenced by: submateq 33333 |
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