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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 10215 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∅ ⊊ (𝐴𝐹𝐵)) |
4 | 1, 2 | genpss 10216 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ⊆ Q) |
5 | vex 3412 | . . . . 5 ⊢ 𝑥 ∈ V | |
6 | vex 3412 | . . . . 5 ⊢ 𝑦 ∈ V | |
7 | genpcl.3 | . . . . 5 ⊢ (ℎ ∈ Q → (𝑓 <Q 𝑔 ↔ (ℎ𝐺𝑓) <Q (ℎ𝐺𝑔))) | |
8 | 5, 6, 7 | caovord 7169 | . . . 4 ⊢ (𝑧 ∈ Q → (𝑥 <Q 𝑦 ↔ (𝑧𝐺𝑥) <Q (𝑧𝐺𝑦))) |
9 | genpcl.4 | . . . 4 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | |
10 | 1, 2, 8, 9 | genpnnp 10217 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ¬ (𝐴𝐹𝐵) = Q) |
11 | dfpss2 3948 | . . 3 ⊢ ((𝐴𝐹𝐵) ⊊ Q ↔ ((𝐴𝐹𝐵) ⊆ Q ∧ ¬ (𝐴𝐹𝐵) = Q)) | |
12 | 4, 10, 11 | sylanbrc 575 | . 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 10218 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
15 | 14 | alrimdv 1888 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)))) |
16 | vex 3412 | . . . . . 6 ⊢ 𝑧 ∈ V | |
17 | vex 3412 | . . . . . 6 ⊢ 𝑤 ∈ V | |
18 | 16, 17, 7 | caovord 7169 | . . . . 5 ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤))) |
19 | 16, 17, 9 | caovcom 7155 | . . . . 5 ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧) |
20 | 1, 2, 18, 19 | genpnmax 10219 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)) |
21 | 15, 20 | jcad 505 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → (∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)) ∧ ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))) |
22 | 21 | ralrimiv 3125 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑓 ∈ (𝐴𝐹𝐵)(∀𝑥(𝑥 <Q 𝑓 → 𝑥 ∈ (𝐴𝐹𝐵)) ∧ ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)) |
23 | elnp 10199 | . 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 1324 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1505 = wceq 1507 ∈ wcel 2048 {cab 2753 ∀wral 3082 ∃wrex 3083 ⊆ wss 3825 ⊊ wpss 3826 ∅c0 4173 class class class wbr 4923 (class class class)co 6970 ∈ cmpo 6972 Qcnq 10064 <Q cltq 10070 Pcnp 10071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-oadd 7901 df-omul 7902 df-er 8081 df-ni 10084 df-mi 10086 df-lti 10087 df-ltpq 10122 df-enq 10123 df-nq 10124 df-ltnq 10130 df-np 10193 |
This theorem is referenced by: addclpr 10230 mulclpr 10232 |
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