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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 1125 | . 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 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑀 ∈ ℤ) → 𝜒) |
15 | 14 | expcom 413 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝜑 → 𝜒)) |
16 | 3anass 1092 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ↔ (𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘))) | |
17 | ancom 460 | . . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) ↔ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) | |
18 | 16, 17 | bitri 275 | . . . . . . . . 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 1169 | . . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝜃) ∧ (𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) ∧ 𝑀 ∈ ℤ) → 𝜏) |
21 | 20 | anasss 466 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜃) ∧ ((𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝑀 ∈ ℤ)) → 𝜏) |
22 | 18, 21 | sylan2b 593 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝜃) ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
23 | 22 | 3impa 1107 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝜃 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → 𝜏) |
24 | 23 | 3com23 1123 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘) ∧ 𝜃) → 𝜏) |
25 | 24 | 3expia 1118 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 𝑀 ≤ 𝑘)) → (𝜃 → 𝜏)) |
26 | 25 | expcom 413 | . . . 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 12655 | . 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 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7404 1c1 11110 + caddc 11112 ≤ cle 11250 ℤcz 12559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 |
This theorem is referenced by: 2ap1caineq 41503 |
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