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Mirrors > Home > ILE Home > Th. List > recexpr | GIF version |
Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) |
Ref | Expression |
---|---|
recexpr | ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq12 3769 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑧 <Q 𝑤 ↔ 𝑢 <Q 𝑣)) | |
2 | simpr 103 | . . . . . . . . 9 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑤 = 𝑣) | |
3 | 2 | fveq2d 5182 | . . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (*Q‘𝑤) = (*Q‘𝑣)) |
4 | 3 | eleq1d 2106 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((*Q‘𝑤) ∈ (2nd ‘𝐴) ↔ (*Q‘𝑣) ∈ (2nd ‘𝐴))) |
5 | 1, 4 | anbi12d 442 | . . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴)))) |
6 | 5 | cbvexdva 1804 | . . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴)))) |
7 | 6 | cbvabv 2161 | . . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴))} |
8 | simpl 102 | . . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑧 = 𝑢) | |
9 | 2, 8 | breq12d 3777 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑤 <Q 𝑧 ↔ 𝑣 <Q 𝑢)) |
10 | 3 | eleq1d 2106 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((*Q‘𝑤) ∈ (1st ‘𝐴) ↔ (*Q‘𝑣) ∈ (1st ‘𝐴))) |
11 | 9, 10 | anbi12d 442 | . . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴)))) |
12 | 11 | cbvexdva 1804 | . . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴)))) |
13 | 12 | cbvabv 2161 | . . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴))} |
14 | 7, 13 | opeq12i 3554 | . . 3 ⊢ 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉 = 〈{𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴))}, {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴))}〉 |
15 | 14 | recexprlempr 6730 | . 2 ⊢ (𝐴 ∈ P → 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉 ∈ P) |
16 | 14 | recexprlemex 6735 | . 2 ⊢ (𝐴 ∈ P → (𝐴 ·P 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉) = 1P) |
17 | oveq2 5520 | . . . 4 ⊢ (𝑥 = 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉 → (𝐴 ·P 𝑥) = (𝐴 ·P 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉)) | |
18 | 17 | eqeq1d 2048 | . . 3 ⊢ (𝑥 = 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉 → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉) = 1P)) |
19 | 18 | rspcev 2656 | . 2 ⊢ ((〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉 ∈ P ∧ (𝐴 ·P 〈{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))}〉) = 1P) → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P) |
20 | 15, 16, 19 | syl2anc 391 | 1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 {cab 2026 ∃wrex 2307 〈cop 3378 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 1st c1st 5765 2nd c2nd 5766 *Qcrq 6382 <Q cltq 6383 Pcnp 6389 1Pc1p 6390 ·P cmp 6392 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-imp 6567 |
This theorem is referenced by: ltmprr 6740 recexgt0sr 6858 |
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