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Mirrors > Home > MPE Home > Th. List > Mathboxes > uzindd | Structured version Visualization version GIF version |
Description: Induction on the upper integers that start at 𝑀. The first four hypotheses give us the substitution instances we need; the following two are the basis and the induction step, a deduction version. (Contributed by metakunt, 8-Jun-2024.) |
Ref | Expression |
---|---|
uzindd.1 | ⊢ (𝑗 = 𝑀 → (𝜓 ↔ 𝜒)) |
uzindd.2 | ⊢ (𝑗 = 𝑘 → (𝜓 ↔ 𝜃)) |
uzindd.3 | ⊢ (𝑗 = (𝑘 + 1) → (𝜓 ↔ 𝜏)) |
uzindd.4 | ⊢ (𝑗 = 𝑁 → (𝜓 ↔ 𝜂)) |
uzindd.5 | ⊢ (𝜑 → 𝜒) |
uzindd.6 | ⊢ ((𝜑 ∧ 𝜃 ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
uzindd.7 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
uzindd.8 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
uzindd.9 | ⊢ (𝜑 → 𝑀 ≤ 𝑁) |
Ref | Expression |
---|---|
uzindd | ⊢ (𝜑 → 𝜂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzindd.7 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | uzindd.8 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | uzindd.9 | . . 3 ⊢ (𝜑 → 𝑀 ≤ 𝑁) | |
4 | 1, 2, 3 | 3jca 1130 | . 2 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
5 | uzindd.1 | . . . 4 ⊢ (𝑗 = 𝑀 → (𝜓 ↔ 𝜒)) | |
6 | 5 | imbi2d 344 | . . 3 ⊢ (𝑗 = 𝑀 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
7 | uzindd.2 | . . . 4 ⊢ (𝑗 = 𝑘 → (𝜓 ↔ 𝜃)) | |
8 | 7 | imbi2d 344 | . . 3 ⊢ (𝑗 = 𝑘 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
9 | uzindd.3 | . . . 4 ⊢ (𝑗 = (𝑘 + 1) → (𝜓 ↔ 𝜏)) | |
10 | 9 | imbi2d 344 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
11 | uzindd.4 | . . . 4 ⊢ (𝑗 = 𝑁 → (𝜓 ↔ 𝜂)) | |
12 | 11 | imbi2d 344 | . . 3 ⊢ (𝑗 = 𝑁 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜂))) |
13 | uzindd.5 | . . . . 5 ⊢ (𝜑 → 𝜒) | |
14 | 13 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝜒) |
15 | 14 | expcom 417 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → 𝜒)) |
16 | 3anass 1097 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ↔ (𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘))) | |
17 | ancom 464 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) ↔ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) | |
18 | 16, 17 | bitri 278 | . . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ↔ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) |
19 | uzindd.6 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜃 ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) | |
20 | 19 | ad4ant123 1174 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜃) ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) ∧ 𝑀 ∈ ℤ) → 𝜏) |
21 | 20 | anasss 470 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜃) ∧ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) → 𝜏) |
22 | 18, 21 | sylan2b 597 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
23 | 22 | 3impa 1112 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜃 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
24 | 23 | 3com23 1128 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝜃) → 𝜏) |
25 | 24 | 3expia 1123 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → (𝜃 → 𝜏)) |
26 | 25 | expcom 417 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → (𝜑 → (𝜃 → 𝜏))) |
27 | 26 | a2d 29 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) → ((𝜑 → 𝜃) → (𝜑 → 𝜏))) |
28 | 6, 8, 10, 12, 15, 27 | uzind 12234 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁) → (𝜑 → 𝜂)) |
29 | 4, 28 | mpcom 38 | 1 ⊢ (𝜑 → 𝜂) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 1c1 10695 + caddc 10697 ≤ cle 10833 ℤcz 12141 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 |
This theorem is referenced by: 2ap1caineq 39770 |
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