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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 3929 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑧 <Q 𝑤 ↔ 𝑢 <Q 𝑣)) | |
2 | simpr 109 | . . . . . . . . 9 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑤 = 𝑣) | |
3 | 2 | fveq2d 5418 | . . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (*Q‘𝑤) = (*Q‘𝑣)) |
4 | 3 | eleq1d 2206 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((*Q‘𝑤) ∈ (2nd ‘𝐴) ↔ (*Q‘𝑣) ∈ (2nd ‘𝐴))) |
5 | 1, 4 | anbi12d 464 | . . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴)))) |
6 | 5 | cbvexdva 1899 | . . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴)))) |
7 | 6 | cbvabv 2262 | . . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q‘𝑤) ∈ (2nd ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q‘𝑣) ∈ (2nd ‘𝐴))} |
8 | simpl 108 | . . . . . . . 8 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → 𝑧 = 𝑢) | |
9 | 2, 8 | breq12d 3937 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → (𝑤 <Q 𝑧 ↔ 𝑣 <Q 𝑢)) |
10 | 3 | eleq1d 2206 | . . . . . . 7 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((*Q‘𝑤) ∈ (1st ‘𝐴) ↔ (*Q‘𝑣) ∈ (1st ‘𝐴))) |
11 | 9, 10 | anbi12d 464 | . . . . . 6 ⊢ ((𝑧 = 𝑢 ∧ 𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴)))) |
12 | 11 | cbvexdva 1899 | . . . . 5 ⊢ (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴)))) |
13 | 12 | cbvabv 2262 | . . . 4 ⊢ {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q‘𝑤) ∈ (1st ‘𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q‘𝑣) ∈ (1st ‘𝐴))} |
14 | 7, 13 | opeq12i 3705 | . . 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 7433 | . 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 7438 | . 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 5775 | . . . 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 2146 | . . 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 2784 | . 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 408 | 1 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2123 ∃wrex 2415 〈cop 3525 class class class wbr 3924 ‘cfv 5118 (class class class)co 5767 1st c1st 6029 2nd c2nd 6030 *Qcrq 7085 <Q cltq 7086 Pcnp 7092 1Pc1p 7093 ·P cmp 7095 |
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 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-i1p 7268 df-imp 7270 |
This theorem is referenced by: ltmprr 7443 recexgt0sr 7574 |
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