| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 10957 | . . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q}) | |
| 2 | cnvimass 6100 | . . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q | |
| 3 | 1, 2 | eqsstri 4030 | . . . . 5 ⊢ *Q ⊆ dom ·Q |
| 4 | mulnqf 10989 | . . . . . 6 ⊢ ·Q :(Q × Q)⟶Q | |
| 5 | 4 | fdmi 6747 | . . . . 5 ⊢ dom ·Q = (Q × Q) |
| 6 | 3, 5 | sseqtri 4032 | . . . 4 ⊢ *Q ⊆ (Q × Q) |
| 7 | dmss 5913 | . . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q)) | |
| 8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom *Q ⊆ dom (Q × Q) |
| 9 | dmxpid 5941 | . . 3 ⊢ dom (Q × Q) = Q | |
| 10 | 8, 9 | sseqtri 4032 | . 2 ⊢ dom *Q ⊆ Q |
| 11 | recclnq 11006 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
| 12 | opelxpi 5722 | . . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) | |
| 13 | 11, 12 | mpdan 687 | . . . . . . 7 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) |
| 14 | df-ov 7434 | . . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) | |
| 15 | recidnq 11005 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
| 16 | 14, 15 | eqtr3id 2791 | . . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q) |
| 17 | ffn 6736 | . . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q)) | |
| 18 | fniniseg 7080 | . . . . . . . 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 2852 | . . . . 5 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) |
| 22 | df-br 5144 | . . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) | |
| 23 | 21, 22 | sylibr 234 | . . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥)) |
| 24 | vex 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 25 | fvex 6919 | . . . . 5 ⊢ (*Q‘𝑥) ∈ V | |
| 26 | 24, 25 | breldm 5919 | . . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q) |
| 27 | 23, 26 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q) |
| 28 | 27 | ssriv 3987 | . 2 ⊢ Q ⊆ dom *Q |
| 29 | 10, 28 | eqssi 4000 | 1 ⊢ dom *Q = Q |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 {csn 4626 〈cop 4632 class class class wbr 5143 × cxp 5683 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Qcnq 10892 1Qc1q 10893 ·Q cmq 10896 *Qcrq 10897 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ni 10912 df-mi 10914 df-lti 10915 df-mpq 10949 df-enq 10951 df-nq 10952 df-erq 10953 df-mq 10955 df-1nq 10956 df-rq 10957 |
| This theorem is referenced by: ltrnq 11019 reclem2pr 11088 |
| Copyright terms: Public domain | W3C validator |