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Mirrors > Home > MPE Home > Th. List > bezoutlem2 | Structured version Visualization version GIF version |
Description: Lemma for bezout 16577. (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 4092 | . . . 4 ⊢ 𝑀 ⊆ ℕ |
4 | nnuz 12919 | . . . 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 16573 | . . . . 5 ⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀)) |
9 | ne0i 4347 | . . . . 5 ⊢ ((abs‘𝐴) ∈ 𝑀 → 𝑀 ≠ ∅) | |
10 | 8, 9 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝑀 ≠ ∅)) |
11 | eqid 2735 | . . . . . . 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 16573 | . . . . . 6 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})) |
13 | rexcom 3288 | . . . . . . . . . 10 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) | |
14 | 6 | zcnd 12721 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
15 | 14 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ) |
16 | zcn 12616 | . . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
17 | 16 | ad2antll 729 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ) |
18 | 15, 17 | mulcld 11279 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ) |
19 | 7 | zcnd 12721 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 19 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ) |
21 | zcn 12616 | . . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
22 | 21 | ad2antrl 728 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ) |
23 | 20, 22 | mulcld 11279 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ) |
24 | 18, 23 | addcomd 11461 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥))) |
25 | 24 | eqeq2d 2746 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
26 | 25 | 2rexbidva 3218 | . . . . . . . . . 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 3441 | . . . . . . . 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 2787 | . . . . . . 7 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}) |
30 | 29 | eleq2d 2825 | . . . . . 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 4347 | . . . . 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 3035 | . . . . 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 860 | . . 3 ⊢ (𝜑 → 𝑀 ≠ ∅) |
38 | infssuzcl 12972 | . . 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 2843 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 ⊆ wss 3963 ∅c0 4339 ‘cfv 6563 (class class class)co 7431 infcinf 9479 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ℕcn 12264 ℤcz 12611 ℤ≥cuz 12876 abscabs 15270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 |
This theorem is referenced by: bezoutlem3 16575 bezoutlem4 16576 |
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