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Mirrors > Home > MPE Home > Th. List > eluzadd | Structured version Visualization version GIF version |
Description: Membership in a later upper set of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by SN, 7-Feb-2025.) |
Ref | Expression |
---|---|
eluzadd | ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12765 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | zaddcl 12540 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 + 𝐾) ∈ ℤ) | |
3 | 1, 2 | sylan 580 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑀 + 𝐾) ∈ ℤ) |
4 | eluzelz 12770 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
5 | zaddcl 12540 | . . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) | |
6 | 4, 5 | sylan 580 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ ℤ) |
7 | 1 | zred 12604 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
8 | 7 | adantr 481 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → 𝑀 ∈ ℝ) |
9 | eluzelre 12771 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → 𝑁 ∈ ℝ) |
11 | zre 12500 | . . . 4 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
12 | 11 | adantl 482 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → 𝐾 ∈ ℝ) |
13 | eluzle 12773 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
14 | 13 | adantr 481 | . . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → 𝑀 ≤ 𝑁) |
15 | 8, 10, 12, 14 | leadd1dd 11766 | . 2 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑀 + 𝐾) ≤ (𝑁 + 𝐾)) |
16 | eluz2 12766 | . 2 ⊢ ((𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾)) ↔ ((𝑀 + 𝐾) ∈ ℤ ∧ (𝑁 + 𝐾) ∈ ℤ ∧ (𝑀 + 𝐾) ≤ (𝑁 + 𝐾))) | |
17 | 3, 6, 15, 16 | syl3anbrc 1343 | 1 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ 𝐾 ∈ ℤ) → (𝑁 + 𝐾) ∈ (ℤ≥‘(𝑀 + 𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5104 ‘cfv 6494 (class class class)co 7354 ℝcr 11047 + caddc 11051 ≤ cle 11187 ℤcz 12496 ℤ≥cuz 12760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-n0 12411 df-z 12497 df-uz 12761 |
This theorem is referenced by: eluzaddi 12791 seqshft2 13931 shftuz 14951 isumshft 15721 vdwlem2 16851 vdwlem8 16857 mulgnndir 18901 efgcpbllemb 19533 plymullem1 25571 coeeulem 25581 ulmshftlem 25744 ulmshft 25745 fsum2dsub 33111 caushft 36209 |
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