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| Description: Lemma for submateq 33809. (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 13596 | . . . . 5 ⊢ (1...𝑁) ⊆ ℕ | |
| 2 | submateqlem1.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
| 3 | 1, 2 | sselid 3980 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ) | 
| 4 | 3 | nnzd 12642 | . . 3 ⊢ (𝜑 → 𝐾 ∈ ℤ) | 
| 5 | submateqlem1.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 6 | 5 | nnzd 12642 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 7 | fz1ssnn 13596 | . . . . 5 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
| 8 | submateqlem1.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
| 9 | 7, 8 | sselid 3980 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ) | 
| 10 | 9 | nnzd 12642 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 11 | submateqlem1.1 | . . 3 ⊢ (𝜑 → 𝐾 ≤ 𝑀) | |
| 12 | 9 | nnred 12282 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | 
| 13 | 5 | nnred 12282 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℝ) | 
| 14 | 1red 11263 | . . . . 5 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 15 | 13, 14 | resubcld 11692 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ) | 
| 16 | elfzle2 13569 | . . . . 5 ⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ≤ (𝑁 − 1)) | |
| 17 | 8, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ≤ (𝑁 − 1)) | 
| 18 | 13 | lem1d 12202 | . . . 4 ⊢ (𝜑 → (𝑁 − 1) ≤ 𝑁) | 
| 19 | 12, 15, 13, 17, 18 | letrd 11419 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) | 
| 20 | 4, 6, 10, 11, 19 | elfzd 13556 | . 2 ⊢ (𝜑 → 𝑀 ∈ (𝐾...𝑁)) | 
| 21 | 1zzd 12650 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 22 | 10 | peano2zd 12727 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ) | 
| 23 | 9 | nnnn0d 12589 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ0) | 
| 24 | 23 | nn0ge0d 12592 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑀) | 
| 25 | 1re 11262 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 26 | addge02 11775 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → 1 ≤ (𝑀 + 1)) | 
| 29 | 5 | nnnn0d 12589 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | 
| 30 | nn0ltlem1 12680 | . . . . . . 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 257 | . . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁) | 
| 33 | nnltp1le 12676 | . . . . . 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 232 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ≤ 𝑁) | 
| 36 | 21, 6, 22, 28, 35 | elfzd 13556 | . . 3 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁)) | 
| 37 | 3 | nnred 12282 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ ℝ) | 
| 38 | nnleltp1 12675 | . . . . . . . 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 232 | . . . . . 6 ⊢ (𝜑 → 𝐾 < (𝑀 + 1)) | 
| 41 | 37, 40 | ltned 11398 | . . . . 5 ⊢ (𝜑 → 𝐾 ≠ (𝑀 + 1)) | 
| 42 | 41 | necomd 2995 | . . . 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 3961 | . 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 206 ∧ wa 395 ∈ wcel 2107 ≠ wne 2939 ∖ cdif 3947 {csn 4625 class class class wbr 5142 (class class class)co 7432 ℝcr 11155 0cc0 11156 1c1 11157 + caddc 11159 < clt 11296 ≤ cle 11297 − cmin 11493 ℕcn 12267 ℕ0cn0 12528 ...cfz 13548 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-fz 13549 | 
| This theorem is referenced by: submateq 33809 | 
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