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Mirrors > Home > ILE Home > Th. List > bezoutlemb | GIF version |
Description: Lemma for Bézout's identity. The is-bezout condition is satisfied by 𝐵. (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 |
---|---|
bezoutlemb | ⊢ (𝜃 → [𝐵 / 𝑟]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 9058 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 1z 9073 | . . 3 ⊢ 1 ∈ ℤ | |
3 | bezoutlema.a | . . . . . . 7 ⊢ (𝜃 → 𝐴 ∈ ℕ0) | |
4 | 3 | nn0cnd 9025 | . . . . . 6 ⊢ (𝜃 → 𝐴 ∈ ℂ) |
5 | 4 | mul01d 8148 | . . . . 5 ⊢ (𝜃 → (𝐴 · 0) = 0) |
6 | 5 | oveq1d 5782 | . . . 4 ⊢ (𝜃 → ((𝐴 · 0) + (𝐵 · 1)) = (0 + (𝐵 · 1))) |
7 | bezoutlema.b | . . . . . . 7 ⊢ (𝜃 → 𝐵 ∈ ℕ0) | |
8 | 7 | nn0cnd 9025 | . . . . . 6 ⊢ (𝜃 → 𝐵 ∈ ℂ) |
9 | 1cnd 7775 | . . . . . 6 ⊢ (𝜃 → 1 ∈ ℂ) | |
10 | 8, 9 | mulcld 7779 | . . . . 5 ⊢ (𝜃 → (𝐵 · 1) ∈ ℂ) |
11 | 10 | addid2d 7905 | . . . 4 ⊢ (𝜃 → (0 + (𝐵 · 1)) = (𝐵 · 1)) |
12 | 8 | mulid1d 7776 | . . . 4 ⊢ (𝜃 → (𝐵 · 1) = 𝐵) |
13 | 6, 11, 12 | 3eqtrrd 2175 | . . 3 ⊢ (𝜃 → 𝐵 = ((𝐴 · 0) + (𝐵 · 1))) |
14 | oveq2 5775 | . . . . . 6 ⊢ (𝑠 = 0 → (𝐴 · 𝑠) = (𝐴 · 0)) | |
15 | 14 | oveq1d 5782 | . . . . 5 ⊢ (𝑠 = 0 → ((𝐴 · 𝑠) + (𝐵 · 𝑡)) = ((𝐴 · 0) + (𝐵 · 𝑡))) |
16 | 15 | eqeq2d 2149 | . . . 4 ⊢ (𝑠 = 0 → (𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 0) + (𝐵 · 𝑡)))) |
17 | oveq2 5775 | . . . . . 6 ⊢ (𝑡 = 1 → (𝐵 · 𝑡) = (𝐵 · 1)) | |
18 | 17 | oveq2d 5783 | . . . . 5 ⊢ (𝑡 = 1 → ((𝐴 · 0) + (𝐵 · 𝑡)) = ((𝐴 · 0) + (𝐵 · 1))) |
19 | 18 | eqeq2d 2149 | . . . 4 ⊢ (𝑡 = 1 → (𝐵 = ((𝐴 · 0) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 0) + (𝐵 · 1)))) |
20 | 16, 19 | rspc2ev 2799 | . . 3 ⊢ ((0 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐵 = ((𝐴 · 0) + (𝐵 · 1))) → ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) |
21 | 1, 2, 13, 20 | mp3an12i 1319 | . 2 ⊢ (𝜃 → ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) |
22 | bezoutlema.is-bezout | . . . . 5 ⊢ (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡))) | |
23 | eqeq1 2144 | . . . . . 6 ⊢ (𝑟 = 𝐵 → (𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) | |
24 | 23 | 2rexbidv 2458 | . . . . 5 ⊢ (𝑟 = 𝐵 → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
25 | 22, 24 | syl5bb 191 | . . . 4 ⊢ (𝑟 = 𝐵 → (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
26 | 25 | sbcieg 2936 | . . 3 ⊢ (𝐵 ∈ ℕ0 → ([𝐵 / 𝑟]𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
27 | 7, 26 | syl 14 | . 2 ⊢ (𝜃 → ([𝐵 / 𝑟]𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
28 | 21, 27 | mpbird 166 | 1 ⊢ (𝜃 → [𝐵 / 𝑟]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ∃wrex 2415 [wsbc 2904 (class class class)co 5767 0cc0 7613 1c1 7614 + caddc 7616 · cmul 7618 ℕ0cn0 8970 ℤcz 9047 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: bezoutlemex 11678 |
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