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Mirrors > Home > MPE Home > Th. List > Mathboxes > ee7.2aOLD | Structured version Visualization version GIF version |
Description: Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as 𝐴 mod 𝐵. Here, just one subtraction step is proved to preserve the gcdOLD. The rec function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ee7.2aOLD | ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nndivsub 33702 | . . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑥 ∈ ℕ) ∧ ((𝐴 / 𝑥) ∈ ℕ ∧ 𝐴 < 𝐵)) → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)) | |
2 | 1 | exp32 421 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑥 ∈ ℕ) → ((𝐴 / 𝑥) ∈ ℕ → (𝐴 < 𝐵 → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)))) |
3 | 2 | com23 86 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝑥 ∈ ℕ) → (𝐴 < 𝐵 → ((𝐴 / 𝑥) ∈ ℕ → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)))) |
4 | 3 | 3expia 1113 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝑥 ∈ ℕ → (𝐴 < 𝐵 → ((𝐴 / 𝑥) ∈ ℕ → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ))))) |
5 | 4 | com23 86 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → (𝑥 ∈ ℕ → ((𝐴 / 𝑥) ∈ ℕ → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ))))) |
6 | 5 | imp 407 | . . . . . . 7 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 < 𝐵) → (𝑥 ∈ ℕ → ((𝐴 / 𝑥) ∈ ℕ → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)))) |
7 | 6 | imp 407 | . . . . . 6 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 < 𝐵) ∧ 𝑥 ∈ ℕ) → ((𝐴 / 𝑥) ∈ ℕ → ((𝐵 / 𝑥) ∈ ℕ ↔ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ))) |
8 | 7 | pm5.32d 577 | . . . . 5 ⊢ ((((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 < 𝐵) ∧ 𝑥 ∈ ℕ) → (((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ) ↔ ((𝐴 / 𝑥) ∈ ℕ ∧ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ))) |
9 | 8 | rabbidva 3476 | . . . 4 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 < 𝐵) → {𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)} = {𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)}) |
10 | 9 | supeq1d 8898 | . . 3 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 < 𝐵) → sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < ) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)}, ℕ, < )) |
11 | df-gcdOLD 33705 | . . 3 ⊢ gcdOLD (𝐴, 𝐵) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ (𝐵 / 𝑥) ∈ ℕ)}, ℕ, < ) | |
12 | df-gcdOLD 33705 | . . 3 ⊢ gcdOLD (𝐴, (𝐵 − 𝐴)) = sup({𝑥 ∈ ℕ ∣ ((𝐴 / 𝑥) ∈ ℕ ∧ ((𝐵 − 𝐴) / 𝑥) ∈ ℕ)}, ℕ, < ) | |
13 | 10, 11, 12 | 3eqtr4g 2878 | . 2 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ 𝐴 < 𝐵) → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴))) |
14 | 13 | ex 413 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 < 𝐵 → gcdOLD (𝐴, 𝐵) = gcdOLD (𝐴, (𝐵 − 𝐴)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 class class class wbr 5057 (class class class)co 7145 supcsup 8892 < clt 10663 − cmin 10858 / cdiv 11285 ℕcn 11626 gcdOLD cgcdOLD 33704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-gcdOLD 33705 |
This theorem is referenced by: (None) |
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