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Mirrors > Home > MPE Home > Th. List > zindd | Structured version Visualization version 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 12212 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → -𝑦 ∈ ℤ) | |
2 | elznn0nn 12190 | . . . . . . 7 ⊢ (-𝑦 ∈ ℤ ↔ (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) | |
3 | 1, 2 | sylib 221 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ))) |
4 | simpr 488 | . . . . . . 7 ⊢ ((-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ) → --𝑦 ∈ ℕ) | |
5 | 4 | orim2i 911 | . . . . . 6 ⊢ ((-𝑦 ∈ ℕ0 ∨ (-𝑦 ∈ ℝ ∧ --𝑦 ∈ ℕ)) → (-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ)) |
6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ)) |
7 | zcn 12181 | . . . . . . . 8 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
8 | 7 | negnegd 11180 | . . . . . . 7 ⊢ (𝑦 ∈ ℤ → --𝑦 = 𝑦) |
9 | 8 | eleq1d 2822 | . . . . . 6 ⊢ (𝑦 ∈ ℤ → (--𝑦 ∈ ℕ ↔ 𝑦 ∈ ℕ)) |
10 | 9 | orbi2d 916 | . . . . 5 ⊢ (𝑦 ∈ ℤ → ((-𝑦 ∈ ℕ0 ∨ --𝑦 ∈ ℕ) ↔ (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ))) |
11 | 6, 10 | mpbid 235 | . . . 4 ⊢ (𝑦 ∈ ℤ → (-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ)) |
12 | zindd.1 | . . . . . . . 8 ⊢ (𝑥 = 0 → (𝜑 ↔ 𝜓)) | |
13 | 12 | imbi2d 344 | . . . . . . 7 ⊢ (𝑥 = 0 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜓))) |
14 | zindd.2 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
15 | 14 | imbi2d 344 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜒))) |
16 | zindd.3 | . . . . . . . 8 ⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜏)) | |
17 | 16 | imbi2d 344 | . . . . . . 7 ⊢ (𝑥 = (𝑦 + 1) → ((𝜁 → 𝜑) ↔ (𝜁 → 𝜏))) |
18 | zindd.4 | . . . . . . . 8 ⊢ (𝑥 = -𝑦 → (𝜑 ↔ 𝜃)) | |
19 | 18 | imbi2d 344 | . . . . . . 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 32 | . . . . . . . 8 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → (𝜒 → 𝜏))) |
23 | 22 | a2d 29 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → ((𝜁 → 𝜒) → (𝜁 → 𝜏))) |
24 | 13, 15, 17, 19, 20, 23 | nn0ind 12272 | . . . . . 6 ⊢ (-𝑦 ∈ ℕ0 → (𝜁 → 𝜃)) |
25 | 24 | com12 32 | . . . . 5 ⊢ (𝜁 → (-𝑦 ∈ ℕ0 → 𝜃)) |
26 | 13, 15, 17, 15, 20, 23 | nn0ind 12272 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ0 → (𝜁 → 𝜒)) |
27 | nnnn0 12097 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ0) | |
28 | 26, 27 | syl11 33 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜒)) |
29 | zindd.8 | . . . . . 6 ⊢ (𝜁 → (𝑦 ∈ ℕ → (𝜒 → 𝜃))) | |
30 | 28, 29 | mpdd 43 | . . . . 5 ⊢ (𝜁 → (𝑦 ∈ ℕ → 𝜃)) |
31 | 25, 30 | jaod 859 | . . . 4 ⊢ (𝜁 → ((-𝑦 ∈ ℕ0 ∨ 𝑦 ∈ ℕ) → 𝜃)) |
32 | 11, 31 | syl5 34 | . . 3 ⊢ (𝜁 → (𝑦 ∈ ℤ → 𝜃)) |
33 | 32 | ralrimiv 3104 | . 2 ⊢ (𝜁 → ∀𝑦 ∈ ℤ 𝜃) |
34 | znegcl 12212 | . . . . 5 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
35 | negeq 11070 | . . . . . . . . 9 ⊢ (𝑦 = -𝑥 → -𝑦 = --𝑥) | |
36 | zcn 12181 | . . . . . . . . . 10 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
37 | 36 | negnegd 11180 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℤ → --𝑥 = 𝑥) |
38 | 35, 37 | sylan9eqr 2800 | . . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → -𝑦 = 𝑥) |
39 | 38 | eqcomd 2743 | . . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → 𝑥 = -𝑦) |
40 | 39, 18 | syl 17 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜑 ↔ 𝜃)) |
41 | 40 | bicomd 226 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 = -𝑥) → (𝜃 ↔ 𝜑)) |
42 | 34, 41 | rspcdv 3529 | . . . 4 ⊢ (𝑥 ∈ ℤ → (∀𝑦 ∈ ℤ 𝜃 → 𝜑)) |
43 | 42 | com12 32 | . . 3 ⊢ (∀𝑦 ∈ ℤ 𝜃 → (𝑥 ∈ ℤ → 𝜑)) |
44 | 43 | ralrimiv 3104 | . 2 ⊢ (∀𝑦 ∈ ℤ 𝜃 → ∀𝑥 ∈ ℤ 𝜑) |
45 | zindd.5 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | |
46 | 45 | rspccv 3534 | . 2 ⊢ (∀𝑥 ∈ ℤ 𝜑 → (𝐴 ∈ ℤ → 𝜂)) |
47 | 33, 44, 46 | 3syl 18 | 1 ⊢ (𝜁 → (𝐴 ∈ ℤ → 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ∀wral 3061 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 -cneg 11063 ℕcn 11830 ℕ0cn0 12090 ℤcz 12176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 |
This theorem is referenced by: efexp 15662 pcexp 16412 mulgaddcom 18515 mulginvcom 18516 mulgneg2 18525 mulgass2 19619 cnfldmulg 20395 clmmulg 23998 xrsmulgzz 31006 |
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