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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 12251 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | breq2 5073 | . . 3 ⊢ (𝑚 = 𝑁 → (𝑀 ≤ 𝑚 ↔ 𝑀 ≤ 𝑁)) | |
3 | eluzelz 12256 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | eluzle 12259 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
5 | 2, 3, 4 | elrabd 3685 | . 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 5073 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑀 ≤ 𝑚 ↔ 𝑀 ≤ 𝑘)) | |
12 | 11 | elrab 3683 | . . . . 5 ⊢ (𝑘 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚} ↔ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) |
13 | eluz2 12252 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) | |
14 | 13 | biimpri 230 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → 𝑘 ∈ (ℤ≥‘𝑀)) |
15 | 14 | 3expb 1116 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
16 | 12, 15 | sylan2b 595 | . . . 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 12079 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚}) → 𝜏) |
20 | 1, 5, 19 | syl2anc 586 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 {crab 3145 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 1c1 10541 + caddc 10543 ≤ cle 10679 ℤcz 11984 ℤ≥cuz 12246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 |
This theorem is referenced by: uzind4ALT 12310 uzind4s 12311 uzind4s2 12312 uzind4i 12313 seqexw 13388 seqcl2 13391 seqshft2 13399 seqsplit 13406 seqf1o 13414 seqid2 13419 clim2prod 15247 fprodabs 15331 fprodefsum 15451 seq1st 15918 1stcelcls 22072 caubl 23914 caublcls 23915 volsuplem 24159 cpnord 24535 bcmono 25856 sseqp1 31657 iprodefisumlem 32976 sdclem2 35021 seqpo 35026 mettrifi 35036 incssnn0 39314 dvgrat 40650 monoordxrv 41764 climsuselem1 41894 smonoord 43538 |
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