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Mirrors > Home > MPE Home > Th. List > uzm1 | Structured version Visualization version GIF version |
Description: Choices for an element of an upper interval of integers. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
uzm1 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ≥‘𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12342 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | 1 | a1d 25 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (¬ 𝑁 = 𝑀 → 𝑀 ∈ ℤ)) |
3 | eluzelz 12347 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | peano2zm 12119 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 − 1) ∈ ℤ) |
6 | 5 | a1d 25 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (¬ 𝑁 = 𝑀 → (𝑁 − 1) ∈ ℤ)) |
7 | df-ne 2936 | . . . . . 6 ⊢ (𝑁 ≠ 𝑀 ↔ ¬ 𝑁 = 𝑀) | |
8 | eluzle 12350 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
9 | 1 | zred 12181 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
10 | eluzelre 12348 | . . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
11 | 9, 10 | ltlend 10876 | . . . . . . . 8 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀 < 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≠ 𝑀))) |
12 | 11 | biimprd 251 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀 ≤ 𝑁 ∧ 𝑁 ≠ 𝑀) → 𝑀 < 𝑁)) |
13 | 8, 12 | mpand 695 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 ≠ 𝑀 → 𝑀 < 𝑁)) |
14 | 7, 13 | syl5bir 246 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (¬ 𝑁 = 𝑀 → 𝑀 < 𝑁)) |
15 | zltlem1 12129 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) | |
16 | 1, 3, 15 | syl2anc 587 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑀 < 𝑁 ↔ 𝑀 ≤ (𝑁 − 1))) |
17 | 14, 16 | sylibd 242 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (¬ 𝑁 = 𝑀 → 𝑀 ≤ (𝑁 − 1))) |
18 | 2, 6, 17 | 3jcad 1130 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (¬ 𝑁 = 𝑀 → (𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 1)))) |
19 | eluz2 12343 | . . 3 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ ∧ 𝑀 ≤ (𝑁 − 1))) | |
20 | 18, 19 | syl6ibr 255 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (¬ 𝑁 = 𝑀 → (𝑁 − 1) ∈ (ℤ≥‘𝑀))) |
21 | 20 | orrd 862 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈ (ℤ≥‘𝑀))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 class class class wbr 5040 ‘cfv 6350 (class class class)co 7183 1c1 10629 < clt 10766 ≤ cle 10767 − cmin 10961 ℤcz 12075 ℤ≥cuz 12337 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-n0 11990 df-z 12076 df-uz 12338 |
This theorem is referenced by: uzp1 12374 fzm1 13091 hashfzo 13895 iserex 15119 ntrivcvg 15358 ntrivcvgtail 15361 mulgfval 18357 |
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