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Mirrors > Home > ILE Home > Th. List > uztrn | GIF version |
Description: Transitive law for sets of upper integers. (Contributed by NM, 20-Sep-2005.) |
Ref | Expression |
---|---|
uztrn | ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 9531 | . . 3 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ∈ ℤ) | |
2 | 1 | adantl 277 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ∈ ℤ) |
3 | eluzelz 9535 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝑀 ∈ ℤ) | |
4 | 3 | adantr 276 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ ℤ) |
5 | eluzle 9538 | . . . 4 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝑁 ≤ 𝐾) | |
6 | 5 | adantl 277 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝐾) |
7 | eluzle 9538 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝐾) → 𝐾 ≤ 𝑀) | |
8 | 7 | adantr 276 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ≤ 𝑀) |
9 | eluzelz 9535 | . . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → 𝐾 ∈ ℤ) | |
10 | 9 | adantl 277 | . . . 4 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝐾 ∈ ℤ) |
11 | zletr 9300 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) | |
12 | 2, 10, 4, 11 | syl3anc 1238 | . . 3 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → ((𝑁 ≤ 𝐾 ∧ 𝐾 ≤ 𝑀) → 𝑁 ≤ 𝑀)) |
13 | 6, 8, 12 | mp2and 433 | . 2 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑁 ≤ 𝑀) |
14 | eluz2 9532 | . 2 ⊢ (𝑀 ∈ (ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ≤ 𝑀)) | |
15 | 2, 4, 13, 14 | syl3anbrc 1181 | 1 ⊢ ((𝑀 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑁)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 class class class wbr 4003 ‘cfv 5216 ≤ cle 7991 ℤcz 9251 ℤ≥cuz 9526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-pre-ltwlin 7923 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fv 5224 df-ov 5877 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-neg 8129 df-z 9252 df-uz 9527 |
This theorem is referenced by: uztrn2 9543 fzsplit2 10047 fzass4 10059 fzss1 10060 fzss2 10061 uzsplit 10089 seq3fveq2 10466 ser3mono 10475 seq3split 10476 seq3f1olemqsumkj 10495 seq3f1olemqsumk 10496 seq3id 10505 seq3id2 10506 seq3z 10508 seq3coll 10817 cvgratgt0 11536 mertenslemi1 11538 zproddc 11582 dvdsfac 11860 |
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