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Mirrors > Home > MPE Home > Th. List > bezoutlem2 | Structured version Visualization version GIF version |
Description: Lemma for bezout 16518. (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 4078 | . . . 4 ⊢ 𝑀 ⊆ ℕ |
4 | nnuz 12895 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
5 | 3, 4 | sseqtri 4016 | . . 3 ⊢ 𝑀 ⊆ (ℤ≥‘1) |
6 | bezout.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
7 | bezout.4 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℤ) | |
8 | 2, 6, 7 | bezoutlem1 16514 | . . . . 5 ⊢ (𝜑 → (𝐴 ≠ 0 → (abs‘𝐴) ∈ 𝑀)) |
9 | ne0i 4335 | . . . . 5 ⊢ ((abs‘𝐴) ∈ 𝑀 → 𝑀 ≠ ∅) | |
10 | 8, 9 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 0 → 𝑀 ≠ ∅)) |
11 | eqid 2728 | . . . . . . 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 16514 | . . . . . 6 ⊢ (𝜑 → (𝐵 ≠ 0 → (abs‘𝐵) ∈ {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))})) |
13 | rexcom 3284 | . . . . . . . . . 10 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦))) | |
14 | 6 | zcnd 12697 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
15 | 14 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐴 ∈ ℂ) |
16 | zcn 12593 | . . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
17 | 16 | ad2antll 728 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑥 ∈ ℂ) |
18 | 15, 17 | mulcld 11264 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐴 · 𝑥) ∈ ℂ) |
19 | 7 | zcnd 12697 | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
20 | 19 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝐵 ∈ ℂ) |
21 | zcn 12593 | . . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
22 | 21 | ad2antrl 727 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → 𝑦 ∈ ℂ) |
23 | 20, 22 | mulcld 11264 | . . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝐵 · 𝑦) ∈ ℂ) |
24 | 18, 23 | addcomd 11446 | . . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → ((𝐴 · 𝑥) + (𝐵 · 𝑦)) = ((𝐵 · 𝑦) + (𝐴 · 𝑥))) |
25 | 24 | eqeq2d 2739 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ)) → (𝑧 = ((𝐴 · 𝑥) + (𝐵 · 𝑦)) ↔ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥)))) |
26 | 25 | 2rexbidva 3214 | . . . . . . . . . 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 3437 | . . . . . . . 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 2780 | . . . . . . 7 ⊢ (𝜑 → 𝑀 = {𝑧 ∈ ℕ ∣ ∃𝑦 ∈ ℤ ∃𝑥 ∈ ℤ 𝑧 = ((𝐵 · 𝑦) + (𝐴 · 𝑥))}) |
30 | 29 | eleq2d 2815 | . . . . . 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 4335 | . . . . 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 3034 | . . . . 5 ⊢ ((𝐴 ≠ 0 ∨ 𝐵 ≠ 0) ↔ ¬ (𝐴 = 0 ∧ 𝐵 = 0)) | |
36 | 34, 35 | sylibr 233 | . . . 4 ⊢ (𝜑 → (𝐴 ≠ 0 ∨ 𝐵 ≠ 0)) |
37 | 10, 33, 36 | mpjaod 859 | . . 3 ⊢ (𝜑 → 𝑀 ≠ ∅) |
38 | infssuzcl 12946 | . . 3 ⊢ ((𝑀 ⊆ (ℤ≥‘1) ∧ 𝑀 ≠ ∅) → inf(𝑀, ℝ, < ) ∈ 𝑀) | |
39 | 5, 37, 38 | sylancr 586 | . 2 ⊢ (𝜑 → inf(𝑀, ℝ, < ) ∈ 𝑀) |
40 | 1, 39 | eqeltrid 2833 | 1 ⊢ (𝜑 → 𝐺 ∈ 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ∃wrex 3067 {crab 3429 ⊆ wss 3947 ∅c0 4323 ‘cfv 6548 (class class class)co 7420 infcinf 9464 ℂcc 11136 ℝcr 11137 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 < clt 11278 ℕcn 12242 ℤcz 12588 ℤ≥cuz 12852 abscabs 15213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 |
This theorem is referenced by: bezoutlem3 16516 bezoutlem4 16517 |
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