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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 10339 | . . . . . 6 ⊢ *Q = (◡ ·Q “ {1Q}) | |
2 | cnvimass 5949 | . . . . . 6 ⊢ (◡ ·Q “ {1Q}) ⊆ dom ·Q | |
3 | 1, 2 | eqsstri 4001 | . . . . 5 ⊢ *Q ⊆ dom ·Q |
4 | mulnqf 10371 | . . . . . 6 ⊢ ·Q :(Q × Q)⟶Q | |
5 | 4 | fdmi 6524 | . . . . 5 ⊢ dom ·Q = (Q × Q) |
6 | 3, 5 | sseqtri 4003 | . . . 4 ⊢ *Q ⊆ (Q × Q) |
7 | dmss 5771 | . . . 4 ⊢ (*Q ⊆ (Q × Q) → dom *Q ⊆ dom (Q × Q)) | |
8 | 6, 7 | ax-mp 5 | . . 3 ⊢ dom *Q ⊆ dom (Q × Q) |
9 | dmxpid 5800 | . . 3 ⊢ dom (Q × Q) = Q | |
10 | 8, 9 | sseqtri 4003 | . 2 ⊢ dom *Q ⊆ Q |
11 | recclnq 10388 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
12 | opelxpi 5592 | . . . . . . . 8 ⊢ ((𝑥 ∈ Q ∧ (*Q‘𝑥) ∈ Q) → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) | |
13 | 11, 12 | mpdan 685 | . . . . . . 7 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (Q × Q)) |
14 | df-ov 7159 | . . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘𝑥)) = ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) | |
15 | recidnq 10387 | . . . . . . . 8 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
16 | 14, 15 | syl5eqr 2870 | . . . . . . 7 ⊢ (𝑥 ∈ Q → ( ·Q ‘〈𝑥, (*Q‘𝑥)〉) = 1Q) |
17 | ffn 6514 | . . . . . . . 8 ⊢ ( ·Q :(Q × Q)⟶Q → ·Q Fn (Q × Q)) | |
18 | fniniseg 6830 | . . . . . . . 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 585 | . . . . . 6 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ (◡ ·Q “ {1Q})) |
21 | 20, 1 | eleqtrrdi 2924 | . . . . 5 ⊢ (𝑥 ∈ Q → 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) |
22 | df-br 5067 | . . . . 5 ⊢ (𝑥*Q(*Q‘𝑥) ↔ 〈𝑥, (*Q‘𝑥)〉 ∈ *Q) | |
23 | 21, 22 | sylibr 236 | . . . 4 ⊢ (𝑥 ∈ Q → 𝑥*Q(*Q‘𝑥)) |
24 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
25 | fvex 6683 | . . . . 5 ⊢ (*Q‘𝑥) ∈ V | |
26 | 24, 25 | breldm 5777 | . . . 4 ⊢ (𝑥*Q(*Q‘𝑥) → 𝑥 ∈ dom *Q) |
27 | 23, 26 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → 𝑥 ∈ dom *Q) |
28 | 27 | ssriv 3971 | . 2 ⊢ Q ⊆ dom *Q |
29 | 10, 28 | eqssi 3983 | 1 ⊢ dom *Q = Q |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 {csn 4567 〈cop 4573 class class class wbr 5066 × cxp 5553 ◡ccnv 5554 dom cdm 5555 “ cima 5558 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 Qcnq 10274 1Qc1q 10275 ·Q cmq 10278 *Qcrq 10279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ni 10294 df-mi 10296 df-lti 10297 df-mpq 10331 df-enq 10333 df-nq 10334 df-erq 10335 df-mq 10337 df-1nq 10338 df-rq 10339 |
This theorem is referenced by: ltrnq 10401 reclem2pr 10470 |
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