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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 8377 | . . 3 ⊢ 1 ∈ ℤ | |
2 | 0z 8362 | . . 3 ⊢ 0 ∈ ℤ | |
3 | bezoutlema.b | . . . . . . 7 ⊢ (𝜃 → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0cnd 8343 | . . . . . 6 ⊢ (𝜃 → 𝐵 ∈ ℂ) |
5 | 4 | mul01d 7497 | . . . . 5 ⊢ (𝜃 → (𝐵 · 0) = 0) |
6 | 5 | oveq2d 5548 | . . . 4 ⊢ (𝜃 → ((𝐴 · 1) + (𝐵 · 0)) = ((𝐴 · 1) + 0)) |
7 | bezoutlema.a | . . . . . . 7 ⊢ (𝜃 → 𝐴 ∈ ℕ0) | |
8 | 7 | nn0cnd 8343 | . . . . . 6 ⊢ (𝜃 → 𝐴 ∈ ℂ) |
9 | 1cnd 7135 | . . . . . 6 ⊢ (𝜃 → 1 ∈ ℂ) | |
10 | 8, 9 | mulcld 7139 | . . . . 5 ⊢ (𝜃 → (𝐴 · 1) ∈ ℂ) |
11 | 10 | addid1d 7257 | . . . 4 ⊢ (𝜃 → ((𝐴 · 1) + 0) = (𝐴 · 1)) |
12 | 8 | mulid1d 7136 | . . . 4 ⊢ (𝜃 → (𝐴 · 1) = 𝐴) |
13 | 6, 11, 12 | 3eqtrrd 2118 | . . 3 ⊢ (𝜃 → 𝐴 = ((𝐴 · 1) + (𝐵 · 0))) |
14 | oveq2 5540 | . . . . . 6 ⊢ (𝑠 = 1 → (𝐴 · 𝑠) = (𝐴 · 1)) | |
15 | 14 | oveq1d 5547 | . . . . 5 ⊢ (𝑠 = 1 → ((𝐴 · 𝑠) + (𝐵 · 𝑡)) = ((𝐴 · 1) + (𝐵 · 𝑡))) |
16 | 15 | eqeq2d 2092 | . . . 4 ⊢ (𝑠 = 1 → (𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 1) + (𝐵 · 𝑡)))) |
17 | oveq2 5540 | . . . . . 6 ⊢ (𝑡 = 0 → (𝐵 · 𝑡) = (𝐵 · 0)) | |
18 | 17 | oveq2d 5548 | . . . . 5 ⊢ (𝑡 = 0 → ((𝐴 · 1) + (𝐵 · 𝑡)) = ((𝐴 · 1) + (𝐵 · 0))) |
19 | 18 | eqeq2d 2092 | . . . 4 ⊢ (𝑡 = 0 → (𝐴 = ((𝐴 · 1) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 1) + (𝐵 · 0)))) |
20 | 16, 19 | rspc2ev 2715 | . . 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 1272 | . 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 2087 | . . . . . 6 ⊢ (𝑟 = 𝐴 → (𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) | |
24 | 23 | 2rexbidv 2391 | . . . . 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 190 | . . . 4 ⊢ (𝑟 = 𝐴 → (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
26 | 25 | sbcieg 2846 | . . 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 165 | 1 ⊢ (𝜃 → [𝐴 / 𝑟]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 [wsbc 2815 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 ℕ0cn0 8288 ℤcz 8351 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: bezoutlemex 10390 |
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