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| Mirrors > Home > MPE Home > Th. List > bezoutlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for bezout 16580. (Contributed by Mario Carneiro, 15-Mar-2014.) ( Revised by AV, 30-Sep-2020.) |
| Ref | Expression |
|---|---|
| bezout.1 | ⊢ 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))} |
| bezout.3 | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
| bezout.4 | ⊢ (𝜑 → 𝐵 ∈ ℤ) |
| bezout.2 | ⊢ 𝐺 = inf(𝑀, ℝ, < ) |
| bezout.5 | ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) |
| Ref | Expression |
|---|---|
| bezoutlem2 | ⊢ (𝜑 → 𝐺 ∈ 𝑀) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | bezout.2 | . 2 ⊢ 𝐺 = inf(𝑀, ℝ, < ) | |
| 2 | bezout.1 | . . . . 5 ⊢ 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))} | |
| 3 | 2 | ssrab3 4082 | . . . 4 ⊢ 𝑀 ⊆ ℕ |
| 4 | nnuz 12921 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
| 5 | 3, 4 | sseqtri 4032 | . . 3 ⊢ 𝑀 ⊆ (ℤ≥‘1) |
| 6 | bezout.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
| 7 | bezout.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
| 8 | 2, 6, 7 | bezoutlem1 16576 | . . . . 5 ⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀)) |
| 9 | ne0i 4341 | . . . . 5 ⊢ ((abs‘𝐴) ∈ 𝑀 → 𝑀 ≠ ∅) | |
| 10 | 8, 9 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝑀 ≠ ∅)) |
| 11 | eqid 2737 | . . . . . . 7 ⊢ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))} = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))} | |
| 12 | 11, 7, 6 | bezoutlem1 16576 | . . . . . 6 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})) |
| 13 | rexcom 3290 | . . . . . . . . . 10 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) | |
| 14 | 6 | zcnd 12723 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 15 | 14 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ) |
| 16 | zcn 12618 | . . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
| 17 | 16 | ad2antll 729 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ) |
| 18 | 15, 17 | mulcld 11281 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ) |
| 19 | 7 | zcnd 12723 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 20 | 19 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ) |
| 21 | zcn 12618 | . . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
| 22 | 21 | ad2antrl 728 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ) |
| 23 | 20, 22 | mulcld 11281 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ) |
| 24 | 18, 23 | addcomd 11463 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥))) |
| 25 | 24 | eqeq2d 2748 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
| 26 | 25 | 2rexbidva 3220 | . . . . . . . . . 10 ⊢ (𝜑 → (∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
| 27 | 13, 26 | bitrid 283 | . . . . . . . . 9 ⊢ (𝜑 → (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
| 28 | 27 | rabbidv 3444 | . . . . . . . 8 ⊢ (𝜑 → {𝑧 ∈ ℕ ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))} = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}) |
| 29 | 2, 28 | eqtrid 2789 | . . . . . . 7 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}) |
| 30 | 29 | eleq2d 2827 | . . . . . 6 ⊢ (𝜑 → ((abs‘𝐵) ∈ 𝑀 ↔ (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})) |
| 31 | 12, 30 | sylibrd 259 | . . . . 5 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ 𝑀)) |
| 32 | ne0i 4341 | . . . . 5 ⊢ ((abs‘𝐵) ∈ 𝑀 → 𝑀 ≠ ∅) | |
| 33 | 31, 32 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐵 ≠ 0 → 𝑀 ≠ ∅)) |
| 34 | bezout.5 | . . . . 5 ⊢ (𝜑 → ¬ (𝐴 = 0 ∧ 𝐵 = 0)) | |
| 35 | neorian 3037 | . . . . 5 ⊢ ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) | |
| 36 | 34, 35 | sylibr 234 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
| 37 | 10, 33, 36 | mpjaod 861 | . . 3 ⊢ (𝜑 → 𝑀 ≠ ∅) |
| 38 | infssuzcl 12974 | . . 3 ⊢ ((𝑀 ⊆ (ℤ≥‘1) ∧ 𝑀 ≠ ∅) → inf(𝑀, ℝ, < ) ∈ 𝑀) | |
| 39 | 5, 37, 38 | sylancr 587 | . 2 ⊢ (𝜑 → inf(𝑀, ℝ, < ) ∈ 𝑀) |
| 40 | 1, 39 | eqeltrid 2845 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝑀) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ℕcn 12266 ℤcz 12613 ℤ≥cuz 12878 abscabs 15273 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 |
| This theorem is referenced by: bezoutlem3 16578 bezoutlem4 16579 |
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