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Mirrors > Home > ILE Home > Th. List > zindd | GIF version |
Description: Principle of Mathematical Induction on all integers, deduction version. The first five hypotheses give the substitutions; the last three are the basis, the induction, and the extension to negative numbers. (Contributed by Paul Chapman, 17-Apr-2009.) (Proof shortened by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
zindd.1 | ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) |
zindd.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
zindd.3 | ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) |
zindd.4 | ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) |
zindd.5 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) |
zindd.6 | ⊢ (𝜁 → 𝜓) |
zindd.7 | ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏))) |
zindd.8 | ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) |
Ref | Expression |
---|---|
zindd | ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znegcl 8276 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
2 | elznn0nn 8259 | . . . . . . 7 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) | |
3 | 1, 2 | sylib 127 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) |
4 | simpr 103 | . . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ) → --𝑦 ∈ ℕ) | |
5 | 4 | orim2i 678 | . . . . . 6 ⊢ ((-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ)) → (-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ)) |
6 | 3, 5 | syl 14 | . . . . 5 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ)) |
7 | zcn 8250 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
8 | 7 | negnegd 7313 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → --𝑦 = 𝑦) |
9 | 8 | eleq1d 2106 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (--𝑦 ∈ ℕ ↔ 𝑦 ∈ ℕ)) |
10 | 9 | orbi2d 704 | . . . . 5 ⊢ (𝑦 ∈ ℤ → ((-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ) ↔ (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ))) |
11 | 6, 10 | mpbid 135 | . . . 4 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ)) |
12 | zindd.1 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
13 | 12 | imbi2d 219 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜓))) |
14 | zindd.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
15 | 14 | imbi2d 219 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜒))) |
16 | zindd.3 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) | |
17 | 16 | imbi2d 219 | . . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜏))) |
18 | zindd.4 | . . . . . . . 8 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) | |
19 | 18 | imbi2d 219 | . . . . . . 7 ⊢ (𝑥 = -𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜃))) |
20 | zindd.6 | . . . . . . 7 ⊢ (𝜁 → 𝜓) | |
21 | zindd.7 | . . . . . . . . 9 ⊢ (𝜁 → (𝑦 ∈ ℕ0 → (𝜒 → 𝜏))) | |
22 | 21 | com12 27 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → (𝜒 → 𝜏))) |
23 | 22 | a2d 23 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → ((𝜁 → 𝜒) → (𝜁 → 𝜏))) |
24 | 13, 15, 17, 19, 20, 23 | nn0ind 8352 | . . . . . 6 ⊢ (-𝑦 ∈ ℕ0 → (𝜁 → 𝜃)) |
25 | 24 | com12 27 | . . . . 5 ⊢ (𝜁 → (-𝑦 ∈ ℕ0 → 𝜃)) |
26 | nnnn0 8188 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
27 | 13, 15, 17, 15, 20, 23 | nn0ind 8352 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → 𝜒)) |
28 | 26, 27 | syl 14 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝜁 → 𝜒)) |
29 | 28 | com12 27 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜒)) |
30 | zindd.8 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) | |
31 | 29, 30 | mpdd 36 | . . . . 5 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜃)) |
32 | 25, 31 | jaod 637 | . . . 4 ⊢ (𝜁 → ((-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ) → 𝜃)) |
33 | 11, 32 | syl5 28 | . . 3 ⊢ (𝜁 → (𝑦 ∈ ℤ → 𝜃)) |
34 | 33 | ralrimiv 2391 | . 2 ⊢ (𝜁 → ∀𝑦 ∈ ℤ 𝜃) |
35 | znegcl 8276 | . . . . 5 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
36 | negeq 7204 | . . . . . . . . 9 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
37 | zcn 8250 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
38 | 37 | negnegd 7313 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) |
39 | 36, 38 | sylan9eqr 2094 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → -𝑦 = 𝑥) |
40 | 39 | eqcomd 2045 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → 𝑥 = -𝑦) |
41 | 40, 18 | syl 14 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜑 ↔ 𝜃)) |
42 | 41 | bicomd 129 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜃 ↔ 𝜑)) |
43 | 35, 42 | rspcdv 2659 | . . . 4 ⊢ (𝑥 ∈ ℤ → (∀𝑦 ∈ ℤ 𝜃 → 𝜑)) |
44 | 43 | com12 27 | . . 3 ⊢ (∀𝑦 ∈ ℤ 𝜃 → (𝑥 ∈ ℤ → 𝜑)) |
45 | 44 | ralrimiv 2391 | . 2 ⊢ (∀𝑦 ∈ ℤ 𝜃 → ∀𝑥 ∈ ℤ 𝜑) |
46 | zindd.5 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | |
47 | 46 | rspccv 2653 | . 2 ⊢ (∀𝑥 ∈ ℤ 𝜑 → (𝐴 ∈ ℤ → 𝜂)) |
48 | 34, 45, 47 | 3syl 17 | 1 ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 = wceq 1243 ∈ wcel 1393 ∀wral 2306 (class class class)co 5512 ℝcr 6888 0cc0 6889 1c1 6890 + caddc 6892 -cneg 7183 ℕcn 7914 ℕ0cn0 8181 ℤcz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 |
This theorem is referenced by: (None) |
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