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Mirrors > Home > ILE Home > Th. List > bezoutlema | GIF version |
Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by 𝐴. (Contributed by Jim Kingdon, 30-Dec-2021.) |
Ref | Expression |
---|---|
bezoutlema.is-bezout | ⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) |
bezoutlema.a | ⊢ (𝜃 → 𝐴 ∈ ℕ0) |
bezoutlema.b | ⊢ (𝜃 → 𝐵 ∈ ℕ0) |
Ref | Expression |
---|---|
bezoutlema | ⊢ (𝜃 → [𝐴 / 𝑟]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9080 | . . 3 ⊢ 1 ∈ ℤ | |
2 | 0z 9065 | . . 3 ⊢ 0 ∈ ℤ | |
3 | bezoutlema.b | . . . . . . 7 ⊢ (𝜃 → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0cnd 9032 | . . . . . 6 ⊢ (𝜃 → 𝐵 ∈ ℂ) |
5 | 4 | mul01d 8155 | . . . . 5 ⊢ (𝜃 → (𝐵 · 0) = 0) |
6 | 5 | oveq2d 5790 | . . . 4 ⊢ (𝜃 → ((𝐴 · 1) + (𝐵 · 0)) = ((𝐴 · 1) + 0)) |
7 | bezoutlema.a | . . . . . . 7 ⊢ (𝜃 → 𝐴 ∈ ℕ0) | |
8 | 7 | nn0cnd 9032 | . . . . . 6 ⊢ (𝜃 → 𝐴 ∈ ℂ) |
9 | 1cnd 7782 | . . . . . 6 ⊢ (𝜃 → 1 ∈ ℂ) | |
10 | 8, 9 | mulcld 7786 | . . . . 5 ⊢ (𝜃 → (𝐴 · 1) ∈ ℂ) |
11 | 10 | addid1d 7911 | . . . 4 ⊢ (𝜃 → ((𝐴 · 1) + 0) = (𝐴 · 1)) |
12 | 8 | mulid1d 7783 | . . . 4 ⊢ (𝜃 → (𝐴 · 1) = 𝐴) |
13 | 6, 11, 12 | 3eqtrrd 2177 | . . 3 ⊢ (𝜃 → 𝐴 = ((𝐴 · 1) + (𝐵 · 0))) |
14 | oveq2 5782 | . . . . . 6 ⊢ (𝑠 = 1 → (𝐴 · 𝑠) = (𝐴 · 1)) | |
15 | 14 | oveq1d 5789 | . . . . 5 ⊢ (𝑠 = 1 → ((𝐴 · 𝑠) + (𝐵 · 𝑡)) = ((𝐴 · 1) + (𝐵 · 𝑡))) |
16 | 15 | eqeq2d 2151 | . . . 4 ⊢ (𝑠 = 1 → (𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 1) + (𝐵 · 𝑡)))) |
17 | oveq2 5782 | . . . . . 6 ⊢ (𝑡 = 0 → (𝐵 · 𝑡) = (𝐵 · 0)) | |
18 | 17 | oveq2d 5790 | . . . . 5 ⊢ (𝑡 = 0 → ((𝐴 · 1) + (𝐵 · 𝑡)) = ((𝐴 · 1) + (𝐵 · 0))) |
19 | 18 | eqeq2d 2151 | . . . 4 ⊢ (𝑡 = 0 → (𝐴 = ((𝐴 · 1) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 1) + (𝐵 · 0)))) |
20 | 16, 19 | rspc2ev 2804 | . . 3 ⊢ ((1 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝐴 = ((𝐴 · 1) + (𝐵 · 0))) → ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) |
21 | 1, 2, 13, 20 | mp3an12i 1319 | . 2 ⊢ (𝜃 → ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) |
22 | bezoutlema.is-bezout | . . . . 5 ⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) | |
23 | eqeq1 2146 | . . . . . 6 ⊢ (𝑟 = 𝐴 → (𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) | |
24 | 23 | 2rexbidv 2460 | . . . . 5 ⊢ (𝑟 = 𝐴 → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
25 | 22, 24 | syl5bb 191 | . . . 4 ⊢ (𝑟 = 𝐴 → (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
26 | 25 | sbcieg 2941 | . . 3 ⊢ (𝐴 ∈ ℕ0 → ([𝐴 / 𝑟]𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
27 | 7, 26 | syl 14 | . 2 ⊢ (𝜃 → ([𝐴 / 𝑟]𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
28 | 21, 27 | mpbird 166 | 1 ⊢ (𝜃 → [𝐴 / 𝑟]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2417 [wsbc 2909 (class class class)co 5774 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 ℕ0cn0 8977 ℤcz 9054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 |
This theorem is referenced by: bezoutlemex 11689 |
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