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Mirrors > Home > MPE Home > Th. List > recrecnq | Structured version Visualization version GIF version |
Description: Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
recrecnq | ⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2fveq3 6674 | . . 3 ⊢ (𝑥 = 𝐴 → (*Q‘(*Q‘𝑥)) = (*Q‘(*Q‘𝐴))) | |
2 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | 1, 2 | eqeq12d 2837 | . 2 ⊢ (𝑥 = 𝐴 → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ (*Q‘(*Q‘𝐴)) = 𝐴)) |
4 | mulcomnq 10374 | . . . 4 ⊢ ((*Q‘𝑥) ·Q 𝑥) = (𝑥 ·Q (*Q‘𝑥)) | |
5 | recidnq 10386 | . . . 4 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
6 | 4, 5 | syl5eq 2868 | . . 3 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 𝑥) = 1Q) |
7 | recclnq 10387 | . . . 4 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
8 | recmulnq 10385 | . . . 4 ⊢ ((*Q‘𝑥) ∈ Q → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ ((*Q‘𝑥) ·Q 𝑥) = 1Q)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ ((*Q‘𝑥) ·Q 𝑥) = 1Q)) |
10 | 6, 9 | mpbird 259 | . 2 ⊢ (𝑥 ∈ Q → (*Q‘(*Q‘𝑥)) = 𝑥) |
11 | 3, 10 | vtoclga 3573 | 1 ⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 Qcnq 10273 1Qc1q 10274 ·Q cmq 10277 *Qcrq 10278 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-omul 8106 df-er 8288 df-ni 10293 df-mi 10295 df-lti 10296 df-mpq 10330 df-enq 10332 df-nq 10333 df-erq 10334 df-mq 10336 df-1nq 10337 df-rq 10338 |
This theorem is referenced by: reclem2pr 10469 |
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