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Mirrors > Home > ILE Home > Th. List > nqprrnd | GIF version |
Description: A cut produced from a rational is rounded. Lemma for nqprlu 6737. (Contributed by Jim Kingdon, 8-Dec-2019.) |
Ref | Expression |
---|---|
nqprrnd | ⊢ (𝐴 ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltbtwnnqq 6605 | . . . . . 6 ⊢ (𝐴 <Q 𝑟 ↔ ∃𝑞 ∈ Q (𝐴 <Q 𝑞 ∧ 𝑞 <Q 𝑟)) | |
2 | ancom 262 | . . . . . . 7 ⊢ ((𝐴 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ↔ (𝑞 <Q 𝑟 ∧ 𝐴 <Q 𝑞)) | |
3 | 2 | rexbii 2373 | . . . . . 6 ⊢ (∃𝑞 ∈ Q (𝐴 <Q 𝑞 ∧ 𝑞 <Q 𝑟) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝐴 <Q 𝑞)) |
4 | 1, 3 | bitri 182 | . . . . 5 ⊢ (𝐴 <Q 𝑟 ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝐴 <Q 𝑞)) |
5 | vex 2604 | . . . . . 6 ⊢ 𝑟 ∈ V | |
6 | breq2 3789 | . . . . . 6 ⊢ (𝑥 = 𝑟 → (𝐴 <Q 𝑥 ↔ 𝐴 <Q 𝑟)) | |
7 | 5, 6 | elab 2738 | . . . . 5 ⊢ (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟) |
8 | vex 2604 | . . . . . . . 8 ⊢ 𝑞 ∈ V | |
9 | breq2 3789 | . . . . . . . 8 ⊢ (𝑥 = 𝑞 → (𝐴 <Q 𝑥 ↔ 𝐴 <Q 𝑞)) | |
10 | 8, 9 | elab 2738 | . . . . . . 7 ⊢ (𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑞) |
11 | 10 | anbi2i 444 | . . . . . 6 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) ↔ (𝑞 <Q 𝑟 ∧ 𝐴 <Q 𝑞)) |
12 | 11 | rexbii 2373 | . . . . 5 ⊢ (∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}) ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝐴 <Q 𝑞)) |
13 | 4, 7, 12 | 3bitr4i 210 | . . . 4 ⊢ (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) |
14 | 13 | rgenw 2418 | . . 3 ⊢ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})) |
15 | 14 | a1i 9 | . 2 ⊢ (𝐴 ∈ Q → ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥}))) |
16 | ltbtwnnqq 6605 | . . . 4 ⊢ (𝑞 <Q 𝐴 ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝐴)) | |
17 | breq1 3788 | . . . . 5 ⊢ (𝑥 = 𝑞 → (𝑥 <Q 𝐴 ↔ 𝑞 <Q 𝐴)) | |
18 | 8, 17 | elab 2738 | . . . 4 ⊢ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴) |
19 | breq1 3788 | . . . . . . 7 ⊢ (𝑥 = 𝑟 → (𝑥 <Q 𝐴 ↔ 𝑟 <Q 𝐴)) | |
20 | 5, 19 | elab 2738 | . . . . . 6 ⊢ (𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ 𝑟 <Q 𝐴) |
21 | 20 | anbi2i 444 | . . . . 5 ⊢ ((𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴}) ↔ (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝐴)) |
22 | 21 | rexbii 2373 | . . . 4 ⊢ (∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴}) ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 <Q 𝐴)) |
23 | 16, 18, 22 | 3bitr4i 210 | . . 3 ⊢ (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) |
24 | 23 | rgenw 2418 | . 2 ⊢ ∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) |
25 | 15, 24 | jctil 305 | 1 ⊢ (𝐴 ∈ Q → (∀𝑞 ∈ Q (𝑞 ∈ {𝑥 ∣ 𝑥 <Q 𝐴} ↔ ∃𝑟 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑟 ∈ {𝑥 ∣ 𝑥 <Q 𝐴})) ∧ ∀𝑟 ∈ Q (𝑟 ∈ {𝑥 ∣ 𝐴 <Q 𝑥} ↔ ∃𝑞 ∈ Q (𝑞 <Q 𝑟 ∧ 𝑞 ∈ {𝑥 ∣ 𝐴 <Q 𝑥})))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 {cab 2067 ∀wral 2348 ∃wrex 2349 class class class wbr 3785 Qcnq 6470 <Q cltq 6475 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 |
This theorem is referenced by: nqprxx 6736 |
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