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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 12249 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
2 | breq2 5070 | . . 3 ⊢ (𝑚 = 𝑁 → (𝑀 ≤ 𝑚 ↔ 𝑀 ≤ 𝑁)) | |
3 | eluzelz 12254 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
4 | eluzle 12257 | . . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑁) | |
5 | 2, 3, 4 | elrabd 3682 | . 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 5070 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑀 ≤ 𝑚 ↔ 𝑀 ≤ 𝑘)) | |
12 | 11 | elrab 3680 | . . . . 5 ⊢ (𝑘 ∈ {𝑚 ∈ ℤ ∣ 𝑀 ≤ 𝑚} ↔ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) |
13 | eluz2 12250 | . . . . . . 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 12077 | . 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 1537 ∈ wcel 2114 {crab 3142 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 1c1 10538 + caddc 10540 ≤ cle 10676 ℤcz 11982 ℤ≥cuz 12244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 |
This theorem is referenced by: uzind4ALT 12308 uzind4s 12309 uzind4s2 12310 uzind4i 12311 seqexw 13386 seqcl2 13389 seqshft2 13397 seqsplit 13404 seqf1o 13412 seqid2 13417 clim2prod 15244 fprodabs 15328 fprodefsum 15448 seq1st 15915 1stcelcls 22069 caubl 23911 caublcls 23912 volsuplem 24156 cpnord 24532 bcmono 25853 sseqp1 31653 iprodefisumlem 32972 sdclem2 35032 seqpo 35037 mettrifi 35047 incssnn0 39328 dvgrat 40664 monoordxrv 41778 climsuselem1 41908 smonoord 43551 |
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