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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 9193 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 1z 9208 | . . 3 ⊢ 1 ∈ ℤ | |
3 | bezoutlema.a | . . . . . . 7 ⊢ (𝜃 → 𝐴 ∈ ℕ0) | |
4 | 3 | nn0cnd 9160 | . . . . . 6 ⊢ (𝜃 → 𝐴 ∈ ℂ) |
5 | 4 | mul01d 8282 | . . . . 5 ⊢ (𝜃 → (𝐴 · 0) = 0) |
6 | 5 | oveq1d 5851 | . . . 4 ⊢ (𝜃 → ((𝐴 · 0) + (𝐵 · 1)) = (0 + (𝐵 · 1))) |
7 | bezoutlema.b | . . . . . . 7 ⊢ (𝜃 → 𝐵 ∈ ℕ0) | |
8 | 7 | nn0cnd 9160 | . . . . . 6 ⊢ (𝜃 → 𝐵 ∈ ℂ) |
9 | 1cnd 7906 | . . . . . 6 ⊢ (𝜃 → 1 ∈ ℂ) | |
10 | 8, 9 | mulcld 7910 | . . . . 5 ⊢ (𝜃 → (𝐵 · 1) ∈ ℂ) |
11 | 10 | addid2d 8039 | . . . 4 ⊢ (𝜃 → (0 + (𝐵 · 1)) = (𝐵 · 1)) |
12 | 8 | mulid1d 7907 | . . . 4 ⊢ (𝜃 → (𝐵 · 1) = 𝐵) |
13 | 6, 11, 12 | 3eqtrrd 2202 | . . 3 ⊢ (𝜃 → 𝐵 = ((𝐴 · 0) + (𝐵 · 1))) |
14 | oveq2 5844 | . . . . . 6 ⊢ (𝑠 = 0 → (𝐴 · 𝑠) = (𝐴 · 0)) | |
15 | 14 | oveq1d 5851 | . . . . 5 ⊢ (𝑠 = 0 → ((𝐴 · 𝑠) + (𝐵 · 𝑡)) = ((𝐴 · 0) + (𝐵 · 𝑡))) |
16 | 15 | eqeq2d 2176 | . . . 4 ⊢ (𝑠 = 0 → (𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 0) + (𝐵 · 𝑡)))) |
17 | oveq2 5844 | . . . . . 6 ⊢ (𝑡 = 1 → (𝐵 · 𝑡) = (𝐵 · 1)) | |
18 | 17 | oveq2d 5852 | . . . . 5 ⊢ (𝑡 = 1 → ((𝐴 · 0) + (𝐵 · 𝑡)) = ((𝐴 · 0) + (𝐵 · 1))) |
19 | 18 | eqeq2d 2176 | . . . 4 ⊢ (𝑡 = 1 → (𝐵 = ((𝐴 · 0) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 0) + (𝐵 · 1)))) |
20 | 16, 19 | rspc2ev 2840 | . . 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 1330 | . 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 2171 | . . . . . 6 ⊢ (𝑟 = 𝐵 → (𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐵 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) | |
24 | 23 | 2rexbidv 2489 | . . . . 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 2978 | . . 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 1342 ∈ wcel 2135 ∃wrex 2443 [wsbc 2946 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 · cmul 7749 ℕ0cn0 9105 ℤcz 9182 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-z 9183 |
This theorem is referenced by: bezoutlemex 11919 |
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