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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 1128 | . 2 ⊢ (𝜑 → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
5 | uzindd.1 | . . . 4 ⊢ (𝑗 = 𝑀 → (𝜓 ↔ 𝜒)) | |
6 | 5 | imbi2d 340 | . . 3 ⊢ (𝑗 = 𝑀 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))) |
7 | uzindd.2 | . . . 4 ⊢ (𝑗 = 𝑘 → (𝜓 ↔ 𝜃)) | |
8 | 7 | imbi2d 340 | . . 3 ⊢ (𝑗 = 𝑘 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜃))) |
9 | uzindd.3 | . . . 4 ⊢ (𝑗 = (𝑘 + 1) → (𝜓 ↔ 𝜏)) | |
10 | 9 | imbi2d 340 | . . 3 ⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜏))) |
11 | uzindd.4 | . . . 4 ⊢ (𝑗 = 𝑁 → (𝜓 ↔ 𝜂)) | |
12 | 11 | imbi2d 340 | . . 3 ⊢ (𝑗 = 𝑁 → ((𝜑 → 𝜓) ↔ (𝜑 → 𝜂))) |
13 | uzindd.5 | . . . . 5 ⊢ (𝜑 → 𝜒) | |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝜒) |
15 | 14 | expcom 414 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → 𝜒)) |
16 | 3anass 1095 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ↔ (𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘))) | |
17 | ancom 461 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) ↔ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) | |
18 | 16, 17 | bitri 274 | . . . . . . . . 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 1172 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜃) ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) ∧ 𝑀 ∈ ℤ) → 𝜏) |
21 | 20 | anasss 467 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜃) ∧ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) → 𝜏) |
22 | 18, 21 | sylan2b 594 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
23 | 22 | 3impa 1110 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜃 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
24 | 23 | 3com23 1126 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝜃) → 𝜏) |
25 | 24 | 3expia 1121 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → (𝜃 → 𝜏)) |
26 | 25 | expcom 414 | . . . 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 12595 | . 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 (class class class)co 7357 1c1 11052 + caddc 11054 ≤ cle 11190 ℤcz 12499 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 |
This theorem is referenced by: 2ap1caineq 40553 |
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