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Mirrors > Home > MPE Home > Th. List > Mathboxes > itcovalt2lem2lem1 | Structured version Visualization version GIF version |
Description: Lemma 1 for itcovalt2lem2 45974. (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 12225 | . . . . 5 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℝ) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℝ) |
3 | 2 | adantr 480 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ∈ ℝ) |
4 | simpr 484 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0) | |
5 | simpr 484 | . . . . . 6 ⊢ ((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℕ0) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ∈ ℕ0) |
7 | 4, 6 | nn0addcld 12280 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 𝐶) ∈ ℕ0) |
8 | 7 | nn0red 12277 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 𝐶) ∈ ℝ) |
9 | nnnn0 12223 | . . . . . 6 ⊢ (𝑌 ∈ ℕ → 𝑌 ∈ ℕ0) | |
10 | 9 | ad2antrr 722 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑌 ∈ ℕ0) |
11 | 7, 10 | nn0mulcld 12281 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((𝑁 + 𝐶) · 𝑌) ∈ ℕ0) |
12 | 11 | nn0red 12277 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → ((𝑁 + 𝐶) · 𝑌) ∈ ℝ) |
13 | nn0ge0 12241 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
14 | 13 | adantl 481 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 0 ≤ 𝑁) |
15 | 6 | nn0red 12277 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ∈ ℝ) |
16 | 4 | nn0red 12277 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℝ) |
17 | 15, 16 | addge02d 11547 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (0 ≤ 𝑁 ↔ 𝐶 ≤ (𝑁 + 𝐶))) |
18 | 14, 17 | mpbid 231 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ≤ (𝑁 + 𝐶)) |
19 | simpll 763 | . . . . 5 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑌 ∈ ℕ) | |
20 | 19 | nnred 11971 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝑌 ∈ ℝ) |
21 | 7 | nn0ge0d 12279 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝑁 + 𝐶)) |
22 | nnge1 11984 | . . . . 5 ⊢ (𝑌 ∈ ℕ → 1 ≤ 𝑌) | |
23 | 22 | ad2antrr 722 | . . . 4 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 1 ≤ 𝑌) |
24 | 8, 20, 21, 23 | lemulge11d 11895 | . . 3 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝑁 + 𝐶) ≤ ((𝑁 + 𝐶) · 𝑌)) |
25 | 3, 8, 12, 18, 24 | letrd 11115 | . 2 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → 𝐶 ≤ ((𝑁 + 𝐶) · 𝑌)) |
26 | nn0sub 12266 | . . 3 ⊢ ((𝐶 ∈ ℕ0 ∧ ((𝑁 + 𝐶) · 𝑌) ∈ ℕ0) → (𝐶 ≤ ((𝑁 + 𝐶) · 𝑌) ↔ (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)) | |
27 | 6, 11, 26 | syl2anc 583 | . 2 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (𝐶 ≤ ((𝑁 + 𝐶) · 𝑌) ↔ (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0)) |
28 | 25, 27 | mpbid 231 | 1 ⊢ (((𝑌 ∈ ℕ ∧ 𝐶 ∈ ℕ0) ∧ 𝑁 ∈ ℕ0) → (((𝑁 + 𝐶) · 𝑌) − 𝐶) ∈ ℕ0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2109 class class class wbr 5078 (class class class)co 7268 ℝcr 10854 0cc0 10855 1c1 10856 + caddc 10858 · cmul 10860 ≤ cle 10994 − cmin 11188 ℕcn 11956 ℕ0cn0 12216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-n0 12217 |
This theorem is referenced by: itcovalt2lem2 45974 |
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