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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem1 | Structured version Visualization version GIF version |
Description: Lemma for submateq 33463. (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 13559 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
2 | submateqlem1.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
3 | 1, 2 | sselid 3971 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) |
4 | 3 | nnzd 12610 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
5 | submateqlem1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnzd 12610 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | fz1ssnn 13559 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
8 | submateqlem1.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
9 | 7, 8 | sselid 3971 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
10 | 9 | nnzd 12610 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
11 | submateqlem1.1 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑀) | |
12 | 9 | nnred 12252 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
13 | 5 | nnred 12252 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
14 | 1red 11240 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
15 | 13, 14 | resubcld 11667 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) |
16 | elfzle2 13532 | . . . . 5 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) | |
17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) |
18 | 13 | lem1d 12172 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ≤ 𝑁) |
19 | 12, 15, 13, 17, 18 | letrd 11396 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
20 | 4, 6, 10, 11, 19 | elfzd 13519 | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾...𝑁)) |
21 | 1zzd 12618 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
22 | 10 | peano2zd 12694 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) |
23 | 9 | nnnn0d 12557 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0) |
24 | 23 | nn0ge0d 12560 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀) |
25 | 1re 11239 | . . . . . 6 ⊢ 1 ∈ ℝ | |
26 | addge02 11750 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (0 ≤ 𝑀 ↔ 1 ≤ (𝑀 + 1))) | |
27 | 25, 12, 26 | sylancr 585 | . . . . 5 ⊢ (𝜑 → (0 ≤ 𝑀 ↔ 1 ≤ (𝑀 + 1))) |
28 | 24, 27 | mpbid 231 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝑀 + 1)) |
29 | 5 | nnnn0d 12557 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
30 | nn0ltlem1 12647 | . . . . . . 7 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | |
31 | 23, 29, 30 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
32 | 17, 31 | mpbird 256 | . . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁) |
33 | nnltp1le 12643 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | |
34 | 9, 5, 33 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) |
35 | 32, 34 | mpbid 231 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≤ 𝑁) |
36 | 21, 6, 22, 28, 35 | elfzd 13519 | . . 3 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁)) |
37 | 3 | nnred 12252 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) |
38 | nnleltp1 12642 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝐾 ≤ 𝑀 ↔ 𝐾 < (𝑀 + 1))) | |
39 | 3, 9, 38 | syl2anc 582 | . . . . . . 7 ⊢ (𝜑 → (𝐾 ≤ 𝑀 ↔ 𝐾 < (𝑀 + 1))) |
40 | 11, 39 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → 𝐾 < (𝑀 + 1)) |
41 | 37, 40 | ltned 11375 | . . . . 5 ⊢ (𝜑 → 𝐾 ≠ (𝑀 + 1)) |
42 | 41 | necomd 2986 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≠ 𝐾) |
43 | nelsn 4665 | . . . 4 ⊢ ((𝑀 + 1) ≠ 𝐾 → ¬ (𝑀 + 1) ∈ {𝐾}) | |
44 | 42, 43 | syl 17 | . . 3 ⊢ (𝜑 → ¬ (𝑀 + 1) ∈ {𝐾}) |
45 | 36, 44 | eldifd 3952 | . 2 ⊢ (𝜑 → (𝑀 + 1) ∈ ((1...𝑁) ∖ {𝐾})) |
46 | 20, 45 | jca 510 | 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 394 ∈ wcel 2098 ≠ wne 2930 ∖ cdif 3938 {csn 4625 class class class wbr 5144 (class class class)co 7413 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 < clt 11273 ≤ cle 11274 − cmin 11469 ℕcn 12237 ℕ0cn0 12497 ...cfz 13511 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 |
This theorem is referenced by: submateq 33463 |
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