![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eluzaddiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of eluzaddi 12878 as of 7-Feb-2025. (Contributed by Paul Chapman, 22-Nov-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eluzsubi.1 | ⊢ 𝑀 ∈ ℤ |
eluzsubi.2 | ⊢ 𝐾 ∈ ℤ |
Ref | Expression |
---|---|
eluzaddiOLD | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 12857 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
2 | eluzsubi.2 | . . 3 ⊢ 𝐾 ∈ ℤ | |
3 | zaddcl 12627 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) | |
4 | 1, 2, 3 | sylancl 584 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ ℤ) |
5 | eluzsubi.1 | . . . 4 ⊢ 𝑀 ∈ ℤ | |
6 | 5 | eluz1i 12855 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
7 | zre 12587 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
8 | 5 | zrei 12589 | . . . . . 6 ⊢ 𝑀 ∈ ℝ |
9 | 2 | zrei 12589 | . . . . . 6 ⊢ 𝐾 ∈ ℝ |
10 | leadd1 11707 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝐾 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ (𝑀 + 𝐾) ≤ (𝑁 + 𝐾))) | |
11 | 8, 9, 10 | mp3an13 1448 | . . . . 5 ⊢ (𝑁 ∈ ℝ → (𝑀 ≤ 𝑁 ↔ (𝑀 + 𝐾) ≤ (𝑁 + 𝐾))) |
12 | 7, 11 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℤ → (𝑀 ≤ 𝑁 ↔ (𝑀 + 𝐾) ≤ (𝑁 + 𝐾))) |
13 | 12 | biimpa 475 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝑀 + 𝐾) ≤ (𝑁 + 𝐾)) |
14 | 6, 13 | sylbi 216 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀 + 𝐾) ≤ (𝑁 + 𝐾)) |
15 | zaddcl 12627 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 + 𝐾) ∈ ℤ) | |
16 | 5, 2, 15 | mp2an 690 | . . 3 ⊢ (𝑀 + 𝐾) ∈ ℤ |
17 | 16 | eluz1i 12855 | . 2 ⊢ ((𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)) ↔ ((𝑁 + 𝐾) ∈ ℤ ∧ (𝑀 + 𝐾) ≤ (𝑁 + 𝐾))) |
18 | 4, 14, 17 | sylanbrc 581 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5144 ‘cfv 6543 (class class class)co 7413 ℝcr 11132 + caddc 11136 ≤ cle 11274 ℤcz 12583 ℤ≥cuz 12847 |
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-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 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |