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Mirrors > Home > MPE Home > Th. List > genpcl | Structured version Visualization version GIF version |
Description: Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
genp.1 | ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) |
genp.2 | ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) |
genpcl.3 | ⊢ (ℎ ∈ Q → (𝑓 <Q 𝑔 ↔ (ℎ𝐺𝑓) <Q (ℎ𝐺𝑔))) |
genpcl.4 | ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) |
genpcl.5 | ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (𝐴𝐹𝐵))) |
Ref | Expression |
---|---|
genpcl | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genp.1 | . . 3 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)}) | |
2 | genp.2 | . . 3 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) | |
3 | 1, 2 | genpn0 11042 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∅ ⊊ (𝐴𝐹𝐵)) |
4 | 1, 2 | genpss 11043 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q) |
5 | vex 3465 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | vex 3465 | . . . . 5 ⊢ 𝑦 ∈ V | |
7 | genpcl.3 | . . . . 5 ⊢ (ℎ ∈ Q → (𝑓 <Q 𝑔 ↔ (ℎ𝐺𝑓) <Q (ℎ𝐺𝑔))) | |
8 | 5, 6, 7 | caovord 7636 | . . . 4 ⊢ (𝑧 ∈ Q → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦))) |
9 | genpcl.4 | . . . 4 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | |
10 | 1, 2, 8, 9 | genpnnp 11044 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝐴𝐹𝐵) = Q) |
11 | dfpss2 4083 | . . 3 ⊢ ((𝐴𝐹𝐵) ⊊ Q ↔ ((𝐴𝐹𝐵) ⊆ Q ∧ ¬ (𝐴𝐹𝐵) = Q)) | |
12 | 4, 10, 11 | sylanbrc 581 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊊ Q) |
13 | genpcl.5 | . . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔𝐺ℎ) → 𝑥 ∈ (𝐴𝐹𝐵))) | |
14 | 1, 2, 13 | genpcd 11045 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
15 | 14 | alrimdv 1924 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
16 | vex 3465 | . . . . . 6 ⊢ 𝑧 ∈ V | |
17 | vex 3465 | . . . . . 6 ⊢ 𝑤 ∈ V | |
18 | 16, 17, 7 | caovord 7636 | . . . . 5 ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤))) |
19 | 16, 17, 9 | caovcom 7622 | . . . . 5 ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧) |
20 | 1, 2, 18, 19 | genpnmax 11046 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)) |
21 | 15, 20 | jcad 511 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)) ∧ ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))) |
22 | 21 | ralrimiv 3134 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑓 ∈ (𝐴𝐹𝐵)(∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)) ∧ ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)) |
23 | elnp 11026 | . 2 ⊢ ((𝐴𝐹𝐵) ∈ P ↔ ((∅ ⊊ (𝐴𝐹𝐵) ∧ (𝐴𝐹𝐵) ⊊ Q) ∧ ∀𝑓 ∈ (𝐴𝐹𝐵)(∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)) ∧ ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))) | |
24 | 3, 12, 22, 23 | syl21anbrc 1341 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1531 = wceq 1533 ∈ wcel 2098 {cab 2702 ∀wral 3050 ∃wrex 3059 ⊆ wss 3946 ⊊ wpss 3947 ∅c0 4324 class class class wbr 5152 (class class class)co 7423 ∈ cmpo 7425 Qcnq 10891 <Q cltq 10897 Pcnp 10898 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-inf2 9680 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7876 df-1st 8002 df-2nd 8003 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-oadd 8499 df-omul 8500 df-er 8733 df-ni 10911 df-mi 10913 df-lti 10914 df-ltpq 10949 df-enq 10950 df-nq 10951 df-ltnq 10957 df-np 11020 |
This theorem is referenced by: addclpr 11057 mulclpr 11059 |
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