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Mirrors > Home > MPE Home > Th. List > uzind4 | Structured version Visualization version GIF version |
Description: Induction on the upper set of integers that starts at an integer 𝑀. The first four hypotheses give us the substitution instances we need, and the last two are the basis and the induction step. (Contributed by NM, 7-Sep-2005.) |
Ref | Expression |
---|---|
uzind4.1 | ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) |
uzind4.2 | ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) |
uzind4.3 | ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) |
uzind4.4 | ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) |
uzind4.5 | ⊢ (𝑀 ∈ ℤ → 𝜓) |
uzind4.6 | ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
uzind4 | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzel2 12236 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | breq2 5034 | . . 3 ⊢ (𝑚 = 𝑁 → (𝑀 ≤ 𝑚 ↔ 𝑀 ≤ 𝑁)) | |
3 | eluzelz 12241 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | eluzle 12244 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
5 | 2, 3, 4 | elrabd 3630 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚}) |
6 | uzind4.1 | . . 3 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) | |
7 | uzind4.2 | . . 3 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) | |
8 | uzind4.3 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
9 | uzind4.4 | . . 3 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) | |
10 | uzind4.5 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝜓) | |
11 | breq2 5034 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑀 ≤ 𝑚 ↔ 𝑀 ≤ 𝑘)) | |
12 | 11 | elrab 3628 | . . . . 5 ⊢ (𝑘 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚} ↔ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) |
13 | eluz2 12237 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) | |
14 | 13 | biimpri 231 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → 𝑘 ∈ (ℤ≥‘𝑀)) |
15 | 14 | 3expb 1117 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
16 | 12, 15 | sylan2b 596 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚}) → 𝑘 ∈ (ℤ≥‘𝑀)) |
17 | uzind4.6 | . . . 4 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚}) → (𝜒 → 𝜃)) |
19 | 6, 7, 8, 9, 10, 18 | uzind3 12064 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚}) → 𝜏) |
20 | 1, 5, 19 | syl2anc 587 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 {crab 3110 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 ≤ cle 10665 ℤcz 11969 ℤ≥cuz 12231 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 |
This theorem is referenced by: uzind4ALT 12295 uzind4s 12296 uzind4s2 12297 uzind4i 12298 seqexw 13380 seqcl2 13384 seqshft2 13392 seqsplit 13399 seqf1o 13407 seqid2 13412 clim2prod 15236 fprodabs 15320 fprodefsum 15440 seq1st 15905 1stcelcls 22066 caubl 23912 caublcls 23913 volsuplem 24159 cpnord 24538 bcmono 25861 sseqp1 31763 iprodefisumlem 33085 sdclem2 35180 seqpo 35185 mettrifi 35195 incssnn0 39652 dvgrat 41016 monoordxrv 42121 climsuselem1 42249 smonoord 43888 |
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