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Mirrors > Home > MPE Home > Th. List > eluzsubiOLD | Structured version Visualization version GIF version |
Description: Obsolete version of eluzsubi 12883 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 |
---|---|
eluzsubiOLD | ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 12860 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑁 ∈ ℤ) | |
2 | eluzsubi.2 | . . 3 ⊢ 𝐾 ∈ ℤ | |
3 | zsubcl 12632 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 − 𝐾) ∈ ℤ) | |
4 | 1, 2, 3 | sylancl 584 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ ℤ) |
5 | eluzsubi.1 | . . . . 5 ⊢ 𝑀 ∈ ℤ | |
6 | zaddcl 12630 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 + 𝐾) ∈ ℤ) | |
7 | 5, 2, 6 | mp2an 690 | . . . 4 ⊢ (𝑀 + 𝐾) ∈ ℤ |
8 | 7 | eluz1i 12858 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) ↔ (𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁)) |
9 | 5 | zrei 12592 | . . . . 5 ⊢ 𝑀 ∈ ℝ |
10 | 2 | zrei 12592 | . . . . 5 ⊢ 𝐾 ∈ ℝ |
11 | zre 12590 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
12 | leaddsub 11718 | . . . . 5 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) | |
13 | 9, 10, 11, 12 | mp3an12i 1461 | . . . 4 ⊢ (𝑁 ∈ ℤ → ((𝑀 + 𝐾) ≤ 𝑁 ↔ 𝑀 ≤ (𝑁 − 𝐾))) |
14 | 13 | biimpa 475 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 + 𝐾) ≤ 𝑁) → 𝑀 ≤ (𝑁 − 𝐾)) |
15 | 8, 14 | sylbi 216 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → 𝑀 ≤ (𝑁 − 𝐾)) |
16 | 5 | eluz1i 12858 | . 2 ⊢ ((𝑁 − 𝐾) ∈ (ℤ≥‘𝑀) ↔ ((𝑁 − 𝐾) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 𝐾))) |
17 | 4, 15, 16 | sylanbrc 581 | 1 ⊢ (𝑁 ∈ (ℤ≥‘(𝑀 + 𝐾)) → (𝑁 − 𝐾) ∈ (ℤ≥‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∈ wcel 2098 class class class wbr 5143 ‘cfv 6542 (class class class)co 7415 ℝcr 11135 + caddc 11139 ≤ cle 11277 − cmin 11472 ℤcz 12586 ℤ≥cuz 12850 |
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 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 |
This theorem is referenced by: (None) |
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