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Mirrors > Home > ILE Home > Th. List > uzind2 | GIF version |
Description: Induction on the upper integers that start after an integer 𝑀. The first four hypotheses give us the substitution instances we need; the last two are the basis and the induction step. (Contributed by NM, 25-Jul-2005.) |
Ref | Expression |
---|---|
uzind2.1 | ⊢ (𝑗 = (𝑀 + 1) → (𝜑 ↔ 𝜓)) |
uzind2.2 | ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) |
uzind2.3 | ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) |
uzind2.4 | ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) |
uzind2.5 | ⊢ (𝑀 ∈ ℤ → 𝜓) |
uzind2.6 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (𝜒 → 𝜃)) |
Ref | Expression |
---|---|
uzind2 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zltp1le 8794 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 ↔ (𝑀 + 1) ≤ 𝑁)) | |
2 | peano2z 8776 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ) | |
3 | uzind2.1 | . . . . . . . . . 10 ⊢ (𝑗 = (𝑀 + 1) → (𝜑 ↔ 𝜓)) | |
4 | 3 | imbi2d 228 | . . . . . . . . 9 ⊢ (𝑗 = (𝑀 + 1) → ((𝑀 ∈ ℤ → 𝜑) ↔ (𝑀 ∈ ℤ → 𝜓))) |
5 | uzind2.2 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) | |
6 | 5 | imbi2d 228 | . . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑀 ∈ ℤ → 𝜑) ↔ (𝑀 ∈ ℤ → 𝜒))) |
7 | uzind2.3 | . . . . . . . . . 10 ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
8 | 7 | imbi2d 228 | . . . . . . . . 9 ⊢ (𝑗 = (𝑘 + 1) → ((𝑀 ∈ ℤ → 𝜑) ↔ (𝑀 ∈ ℤ → 𝜃))) |
9 | uzind2.4 | . . . . . . . . . 10 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) | |
10 | 9 | imbi2d 228 | . . . . . . . . 9 ⊢ (𝑗 = 𝑁 → ((𝑀 ∈ ℤ → 𝜑) ↔ (𝑀 ∈ ℤ → 𝜏))) |
11 | uzind2.5 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝜓) | |
12 | 11 | a1i 9 | . . . . . . . . 9 ⊢ ((𝑀 + 1) ∈ ℤ → (𝑀 ∈ ℤ → 𝜓)) |
13 | zltp1le 8794 | . . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑀 < 𝑘 ↔ (𝑀 + 1) ≤ 𝑘)) | |
14 | uzind2.6 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 < 𝑘) → (𝜒 → 𝜃)) | |
15 | 14 | 3expia 1145 | . . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑀 < 𝑘 → (𝜒 → 𝜃))) |
16 | 13, 15 | sylbird 168 | . . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑘 → (𝜒 → 𝜃))) |
17 | 16 | ex 113 | . . . . . . . . . . . . 13 ⊢ (𝑀 ∈ ℤ → (𝑘 ∈ ℤ → ((𝑀 + 1) ≤ 𝑘 → (𝜒 → 𝜃)))) |
18 | 17 | com3l 80 | . . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤ → ((𝑀 + 1) ≤ 𝑘 → (𝑀 ∈ ℤ → (𝜒 → 𝜃)))) |
19 | 18 | imp 122 | . . . . . . . . . . 11 ⊢ ((𝑘 ∈ ℤ ∧ (𝑀 + 1) ≤ 𝑘) → (𝑀 ∈ ℤ → (𝜒 → 𝜃))) |
20 | 19 | 3adant1 961 | . . . . . . . . . 10 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝑀 + 1) ≤ 𝑘) → (𝑀 ∈ ℤ → (𝜒 → 𝜃))) |
21 | 20 | a2d 26 | . . . . . . . . 9 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ (𝑀 + 1) ≤ 𝑘) → ((𝑀 ∈ ℤ → 𝜒) → (𝑀 ∈ ℤ → 𝜃))) |
22 | 4, 6, 8, 10, 12, 21 | uzind 8847 | . . . . . . . 8 ⊢ (((𝑀 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 + 1) ≤ 𝑁) → (𝑀 ∈ ℤ → 𝜏)) |
23 | 22 | 3exp 1142 | . . . . . . 7 ⊢ ((𝑀 + 1) ∈ ℤ → (𝑁 ∈ ℤ → ((𝑀 + 1) ≤ 𝑁 → (𝑀 ∈ ℤ → 𝜏)))) |
24 | 2, 23 | syl 14 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → ((𝑀 + 1) ≤ 𝑁 → (𝑀 ∈ ℤ → 𝜏)))) |
25 | 24 | com34 82 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → (𝑀 ∈ ℤ → ((𝑀 + 1) ≤ 𝑁 → 𝜏)))) |
26 | 25 | pm2.43a 50 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → ((𝑀 + 1) ≤ 𝑁 → 𝜏))) |
27 | 26 | imp 122 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 + 1) ≤ 𝑁 → 𝜏)) |
28 | 1, 27 | sylbid 148 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 < 𝑁 → 𝜏)) |
29 | 28 | 3impia 1140 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 class class class wbr 3843 (class class class)co 5644 1c1 7341 + caddc 7343 < clt 7512 ≤ cle 7513 ℤcz 8740 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3955 ax-pow 4007 ax-pr 4034 ax-un 4258 ax-setind 4351 ax-cnex 7426 ax-resscn 7427 ax-1cn 7428 ax-1re 7429 ax-icn 7430 ax-addcl 7431 ax-addrcl 7432 ax-mulcl 7433 ax-addcom 7435 ax-addass 7437 ax-distr 7439 ax-i2m1 7440 ax-0lt1 7441 ax-0id 7443 ax-rnegex 7444 ax-cnre 7446 ax-pre-ltirr 7447 ax-pre-ltwlin 7448 ax-pre-lttrn 7449 ax-pre-ltadd 7451 |
This theorem depends on definitions: df-bi 115 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3429 df-sn 3450 df-pr 3451 df-op 3453 df-uni 3652 df-int 3687 df-br 3844 df-opab 3898 df-id 4118 df-xp 4442 df-rel 4443 df-cnv 4444 df-co 4445 df-dm 4446 df-iota 4975 df-fun 5012 df-fv 5018 df-riota 5600 df-ov 5647 df-oprab 5648 df-mpt2 5649 df-pnf 7514 df-mnf 7515 df-xr 7516 df-ltxr 7517 df-le 7518 df-sub 7645 df-neg 7646 df-inn 8413 df-n0 8664 df-z 8741 |
This theorem is referenced by: (None) |
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