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Mirrors > Home > MPE Home > Th. List > dmrecnq | Structured version Visualization version GIF version |
Description: Domain of reciprocal on positive fractions. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmrecnq | ⊢ dom *Q = Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rq 10327 | . . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q}) | |
2 | cnvimass 5942 | . . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q | |
3 | 1, 2 | eqsstri 3998 | . . . . 5 ⊢ *Q ⊆ dom ·Q |
4 | mulnqf 10359 | . . . . . 6 ⊢ ·Q :(Q × Q)⟶Q | |
5 | 4 | fdmi 6517 | . . . . 5 ⊢ dom ·Q = (Q × Q) |
6 | 3, 5 | sseqtri 4000 | . . . 4 ⊢ *Q ⊆ (Q × Q) |
7 | dmss 5764 | . . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom *Q ⊆ dom (Q × Q) |
9 | dmxpid 5793 | . . 3 ⊢ dom (Q × Q) = Q | |
10 | 8, 9 | sseqtri 4000 | . 2 ⊢ dom *Q ⊆ Q |
11 | recclnq 10376 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
12 | opelxpi 5585 | . . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) | |
13 | 11, 12 | mpdan 683 | . . . . . . 7 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) |
14 | df-ov 7148 | . . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) | |
15 | recidnq 10375 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
16 | 14, 15 | syl5eqr 2867 | . . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q) |
17 | ffn 6507 | . . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q)) | |
18 | fniniseg 6822 | . . . . . . . 8 ⊢ ( ·Q Fn (Q × Q) → (〈𝑥, (*Q‘𝑥)〉 ∈ (◡ ·Q “ {1Q}) ↔ (〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q) ∧ ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q))) | |
19 | 4, 17, 18 | mp2b 10 | . . . . . . 7 ⊢ (〈𝑥, (*Q‘𝑥)〉 ∈ (◡ ·Q “ {1Q}) ↔ (〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q) ∧ ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q)) |
20 | 13, 16, 19 | sylanbrc 583 | . . . . . 6 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (◡ ·Q “ {1Q})) |
21 | 20, 1 | eleqtrrdi 2921 | . . . . 5 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) |
22 | df-br 5058 | . . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) | |
23 | 21, 22 | sylibr 235 | . . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥)) |
24 | vex 3495 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | fvex 6676 | . . . . 5 ⊢ (*Q‘𝑥) ∈ V | |
26 | 24, 25 | breldm 5770 | . . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q) |
27 | 23, 26 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q) |
28 | 27 | ssriv 3968 | . 2 ⊢ Q ⊆ dom *Q |
29 | 10, 28 | eqssi 3980 | 1 ⊢ dom *Q = Q |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 〈cop 4563 class class class wbr 5057 × cxp 5546 ◡ccnv 5547 dom cdm 5548 “ cima 5551 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Qcnq 10262 1Qc1q 10263 ·Q cmq 10266 *Qcrq 10267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-omul 8096 df-er 8278 df-ni 10282 df-mi 10284 df-lti 10285 df-mpq 10319 df-enq 10321 df-nq 10322 df-erq 10323 df-mq 10325 df-1nq 10326 df-rq 10327 |
This theorem is referenced by: ltrnq 10389 reclem2pr 10458 |
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