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Mirrors > Home > ILE Home > Th. List > suplocsr | GIF version |
Description: An inhabited, bounded, located set of signed reals has a supremum. (Contributed by Jim Kingdon, 22-Jan-2024.) |
Ref | Expression |
---|---|
suplocsr.m | ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) |
suplocsr.ub | ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) |
suplocsr.loc | ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) |
Ref | Expression |
---|---|
suplocsr | ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suplocsr.m | . . 3 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴) | |
2 | eleq1w 2200 | . . . 4 ⊢ (𝑥 = 𝑎 → (𝑥 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴)) | |
3 | 2 | cbvexv 1890 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑎 𝑎 ∈ 𝐴) |
4 | 1, 3 | sylib 121 | . 2 ⊢ (𝜑 → ∃𝑎 𝑎 ∈ 𝐴) |
5 | opeq1 3705 | . . . . . . 7 ⊢ (𝑏 = 𝑐 → 〈𝑏, 1P〉 = 〈𝑐, 1P〉) | |
6 | 5 | eceq1d 6465 | . . . . . 6 ⊢ (𝑏 = 𝑐 → [〈𝑏, 1P〉] ~R = [〈𝑐, 1P〉] ~R ) |
7 | 6 | oveq2d 5790 | . . . . 5 ⊢ (𝑏 = 𝑐 → (𝑎 +R [〈𝑏, 1P〉] ~R ) = (𝑎 +R [〈𝑐, 1P〉] ~R )) |
8 | 7 | eleq1d 2208 | . . . 4 ⊢ (𝑏 = 𝑐 → ((𝑎 +R [〈𝑏, 1P〉] ~R ) ∈ 𝐴 ↔ (𝑎 +R [〈𝑐, 1P〉] ~R ) ∈ 𝐴)) |
9 | 8 | cbvrabv 2685 | . . 3 ⊢ {𝑏 ∈ P ∣ (𝑎 +R [〈𝑏, 1P〉] ~R ) ∈ 𝐴} = {𝑐 ∈ P ∣ (𝑎 +R [〈𝑐, 1P〉] ~R ) ∈ 𝐴} |
10 | suplocsr.ub | . . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) | |
11 | ltrelsr 7546 | . . . . . . . . . 10 ⊢ <R ⊆ (R × R) | |
12 | 11 | brel 4591 | . . . . . . . . 9 ⊢ (𝑦 <R 𝑥 → (𝑦 ∈ R ∧ 𝑥 ∈ R)) |
13 | 12 | simpld 111 | . . . . . . . 8 ⊢ (𝑦 <R 𝑥 → 𝑦 ∈ R) |
14 | 13 | ralimi 2495 | . . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → ∀𝑦 ∈ 𝐴 𝑦 ∈ R) |
15 | dfss3 3087 | . . . . . . 7 ⊢ (𝐴 ⊆ R ↔ ∀𝑦 ∈ 𝐴 𝑦 ∈ R) | |
16 | 14, 15 | sylibr 133 | . . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → 𝐴 ⊆ R) |
17 | 16 | rexlimivw 2545 | . . . . 5 ⊢ (∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥 → 𝐴 ⊆ R) |
18 | 10, 17 | syl 14 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ R) |
19 | 18 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐴 ⊆ R) |
20 | simpr 109 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴) | |
21 | 10 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) |
22 | suplocsr.loc | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) | |
23 | 22 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ R ∀𝑦 ∈ R (𝑥 <R 𝑦 → (∃𝑧 ∈ 𝐴 𝑥 <R 𝑧 ∨ ∀𝑧 ∈ 𝐴 𝑧 <R 𝑦))) |
24 | 9, 19, 20, 21, 23 | suplocsrlem 7616 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) |
25 | 4, 24 | exlimddv 1870 | 1 ⊢ (𝜑 → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 ∃wex 1468 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 {crab 2420 ⊆ wss 3071 〈cop 3530 class class class wbr 3929 (class class class)co 5774 [cec 6427 Pcnp 7099 1Pc1p 7100 ~R cer 7104 Rcnr 7105 +R cplr 7109 <R cltr 7111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-imp 7277 df-iltp 7278 df-enr 7534 df-nr 7535 df-plr 7536 df-mr 7537 df-ltr 7538 df-0r 7539 df-1r 7540 df-m1r 7541 |
This theorem is referenced by: axpre-suploclemres 7709 |
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