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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 9310 | . . 3 ⊢ 1 ∈ ℤ | |
2 | 0z 9295 | . . 3 ⊢ 0 ∈ ℤ | |
3 | bezoutlema.b | . . . . . . 7 ⊢ (𝜃 → 𝐵 ∈ ℕ0) | |
4 | 3 | nn0cnd 9262 | . . . . . 6 ⊢ (𝜃 → 𝐵 ∈ ℂ) |
5 | 4 | mul01d 8381 | . . . . 5 ⊢ (𝜃 → (𝐵 · 0) = 0) |
6 | 5 | oveq2d 5913 | . . . 4 ⊢ (𝜃 → ((𝐴 · 1) + (𝐵 · 0)) = ((𝐴 · 1) + 0)) |
7 | bezoutlema.a | . . . . . . 7 ⊢ (𝜃 → 𝐴 ∈ ℕ0) | |
8 | 7 | nn0cnd 9262 | . . . . . 6 ⊢ (𝜃 → 𝐴 ∈ ℂ) |
9 | 1cnd 8004 | . . . . . 6 ⊢ (𝜃 → 1 ∈ ℂ) | |
10 | 8, 9 | mulcld 8009 | . . . . 5 ⊢ (𝜃 → (𝐴 · 1) ∈ ℂ) |
11 | 10 | addridd 8137 | . . . 4 ⊢ (𝜃 → ((𝐴 · 1) + 0) = (𝐴 · 1)) |
12 | 8 | mulridd 8005 | . . . 4 ⊢ (𝜃 → (𝐴 · 1) = 𝐴) |
13 | 6, 11, 12 | 3eqtrrd 2227 | . . 3 ⊢ (𝜃 → 𝐴 = ((𝐴 · 1) + (𝐵 · 0))) |
14 | oveq2 5905 | . . . . . 6 ⊢ (𝑠 = 1 → (𝐴 · 𝑠) = (𝐴 · 1)) | |
15 | 14 | oveq1d 5912 | . . . . 5 ⊢ (𝑠 = 1 → ((𝐴 · 𝑠) + (𝐵 · 𝑡)) = ((𝐴 · 1) + (𝐵 · 𝑡))) |
16 | 15 | eqeq2d 2201 | . . . 4 ⊢ (𝑠 = 1 → (𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 1) + (𝐵 · 𝑡)))) |
17 | oveq2 5905 | . . . . . 6 ⊢ (𝑡 = 0 → (𝐵 · 𝑡) = (𝐵 · 0)) | |
18 | 17 | oveq2d 5913 | . . . . 5 ⊢ (𝑡 = 0 → ((𝐴 · 1) + (𝐵 · 𝑡)) = ((𝐴 · 1) + (𝐵 · 0))) |
19 | 18 | eqeq2d 2201 | . . . 4 ⊢ (𝑡 = 0 → (𝐴 = ((𝐴 · 1) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 1) + (𝐵 · 0)))) |
20 | 16, 19 | rspc2ev 2871 | . . 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 1352 | . 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 2196 | . . . . . 6 ⊢ (𝑟 = 𝐴 → (𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) | |
24 | 23 | 2rexbidv 2515 | . . . . 5 ⊢ (𝑟 = 𝐴 → (∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝑟 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)) ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
25 | 22, 24 | bitrid 192 | . . . 4 ⊢ (𝑟 = 𝐴 → (𝜑 ↔ ∃𝑠 ∈ ℤ ∃𝑡 ∈ ℤ 𝐴 = ((𝐴 · 𝑠) + (𝐵 · 𝑡)))) |
26 | 25 | sbcieg 3010 | . . 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 167 | 1 ⊢ (𝜃 → [𝐴 / 𝑟]𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2160 ∃wrex 2469 [wsbc 2977 (class class class)co 5897 0cc0 7842 1c1 7843 + caddc 7845 · cmul 7847 ℕ0cn0 9207 ℤcz 9284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-inn 8951 df-n0 9208 df-z 9285 |
This theorem is referenced by: bezoutlemex 12037 |
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