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Mirrors > Home > MPE Home > Th. List > Mathboxes > itcovalt2lem2lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for itcovalt2lem2 45440. (Contributed by AV, 6-May-2024.) |
Ref | Expression |
---|---|
itcovalt2lem2lem1 | ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0re 11928 | . . . . 5 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℝ) | |
2 | 1 | adantl 486 | . . . 4 ⊢ ((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℝ) |
3 | 2 | adantr 485 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ∈ ℝ) |
4 | simpr 489 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
5 | simpr 489 | . . . . . 6 ⊢ ((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℕ0) | |
6 | 5 | adantr 485 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ∈ ℕ0) |
7 | 4, 6 | nn0addcld 11983 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 𝐶) ∈ ℕ0) |
8 | 7 | nn0red 11980 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 𝐶) ∈ ℝ) |
9 | nnnn0 11926 | . . . . . 6 ⊢ (𝑌 ∈ ℕ → 𝑌 ∈ ℕ0) | |
10 | 9 | ad2antrr 726 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑌 ∈ ℕ0) |
11 | 7, 10 | nn0mulcld 11984 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((𝑁 + 𝐶) · 𝑌) ∈ ℕ0) |
12 | 11 | nn0red 11980 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((𝑁 + 𝐶) · 𝑌) ∈ ℝ) |
13 | nn0ge0 11944 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
14 | 13 | adantl 486 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 0 ≤ 𝑁) |
15 | 6 | nn0red 11980 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ∈ ℝ) |
16 | 4 | nn0red 11980 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
17 | 15, 16 | addge02d 11252 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (0 ≤ 𝑁 ↔ 𝐶 ≤ (𝑁 + 𝐶))) |
18 | 14, 17 | mpbid 235 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ≤ (𝑁 + 𝐶)) |
19 | simpll 767 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑌 ∈ ℕ) | |
20 | 19 | nnred 11674 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑌 ∈ ℝ) |
21 | 7 | nn0ge0d 11982 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝑁 + 𝐶)) |
22 | nnge1 11687 | . . . . 5 ⊢ (𝑌 ∈ ℕ → 1 ≤ 𝑌) | |
23 | 22 | ad2antrr 726 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 1 ≤ 𝑌) |
24 | 8, 20, 21, 23 | lemulge11d 11600 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 𝐶) ≤ ((𝑁 + 𝐶) · 𝑌)) |
25 | 3, 8, 12, 18, 24 | letrd 10820 | . 2 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ≤ ((𝑁 + 𝐶) · 𝑌)) |
26 | nn0sub 11969 | . . 3 ⊢ ((𝐶 ∈ ℕ0 ∧ ((𝑁 + 𝐶) · 𝑌) ∈ ℕ0) → (𝐶 ≤ ((𝑁 + 𝐶) · 𝑌) ↔ (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)) | |
27 | 6, 11, 26 | syl2anc 588 | . 2 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐶 ≤ ((𝑁 + 𝐶) · 𝑌) ↔ (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)) |
28 | 25, 27 | mpbid 235 | 1 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∈ wcel 2112 class class class wbr 5025 (class class class)co 7143 ℝcr 10559 0cc0 10560 1c1 10561 + caddc 10563 · cmul 10565 ≤ cle 10699 − cmin 10893 ℕcn 11659 ℕ0cn0 11919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 |
This theorem is referenced by: itcovalt2lem2 45440 |
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