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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 8917 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 1z 8932 | . . 3 ⊢ 1 ∈ ℤ | |
3 | bezoutlema.a | . . . . . . 7 ⊢ (𝜃 → 𝐴 ∈ ℕ0) | |
4 | 3 | nn0cnd 8884 | . . . . . 6 ⊢ (𝜃 → 𝐴 ∈ ℂ) |
5 | 4 | mul01d 8022 | . . . . 5 ⊢ (𝜃 → (𝐴 · 0) = 0) |
6 | 5 | oveq1d 5721 | . . . 4 ⊢ (𝜃 → ((𝐴 · 0) + (𝐵 · 1)) = (0 + (𝐵 · 1))) |
7 | bezoutlema.b | . . . . . . 7 ⊢ (𝜃 → 𝐵 ∈ ℕ0) | |
8 | 7 | nn0cnd 8884 | . . . . . 6 ⊢ (𝜃 → 𝐵 ∈ ℂ) |
9 | 1cnd 7654 | . . . . . 6 ⊢ (𝜃 → 1 ∈ ℂ) | |
10 | 8, 9 | mulcld 7658 | . . . . 5 ⊢ (𝜃 → (𝐵 · 1) ∈ ℂ) |
11 | 10 | addid2d 7783 | . . . 4 ⊢ (𝜃 → (0 + (𝐵 · 1)) = (𝐵 · 1)) |
12 | 8 | mulid1d 7655 | . . . 4 ⊢ (𝜃 → (𝐵 · 1) = 𝐵) |
13 | 6, 11, 12 | 3eqtrrd 2137 | . . 3 ⊢ (𝜃 → 𝐵 = ((𝐴 · 0) + (𝐵 · 1))) |
14 | oveq2 5714 | . . . . . 6 ⊢ (𝑠 = 0 → (𝐴 · 𝑠) = (𝐴 · 0)) | |
15 | 14 | oveq1d 5721 | . . . . 5 ⊢ (𝑠 = 0 → ((𝐴 · 𝑠) + (𝐵 · 𝑡)) = ((𝐴 · 0) + (𝐵 · 𝑡))) |
16 | 15 | eqeq2d 2111 | . . . 4 ⊢ (𝑠 = 0 → (𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 0) + (𝐵 · 𝑡)))) |
17 | oveq2 5714 | . . . . . 6 ⊢ (𝑡 = 1 → (𝐵 · 𝑡) = (𝐵 · 1)) | |
18 | 17 | oveq2d 5722 | . . . . 5 ⊢ (𝑡 = 1 → ((𝐴 · 0) + (𝐵 · 𝑡)) = ((𝐴 · 0) + (𝐵 · 1))) |
19 | 18 | eqeq2d 2111 | . . . 4 ⊢ (𝑡 = 1 → (𝐵 = ((𝐴 · 0) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 0) + (𝐵 · 1)))) |
20 | 16, 19 | rspc2ev 2758 | . . 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 1287 | . 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 2106 | . . . . . 6 ⊢ (𝑟 = 𝐵 → (𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) | |
24 | 23 | 2rexbidv 2419 | . . . . 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 2893 | . . 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 1299 ∈ wcel 1448 ∃wrex 2376 [wsbc 2862 (class class class)co 5706 0cc0 7500 1c1 7501 + caddc 7503 · cmul 7505 ℕ0cn0 8829 ℤcz 8906 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-ltadd 7611 |
This theorem depends on definitions: df-bi 116 df-3or 931 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-inn 8579 df-n0 8830 df-z 8907 |
This theorem is referenced by: bezoutlemex 11482 |
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