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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 6911 | . . 3 ⊢ (𝑥 = 𝐴 → (*Q‘(*Q‘𝑥)) = (*Q‘(*Q‘𝐴))) | |
| 2 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 3 | 1, 2 | eqeq12d 2753 | . 2 ⊢ (𝑥 = 𝐴 → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ (*Q‘(*Q‘𝐴)) = 𝐴)) | 
| 4 | mulcomnq 10993 | . . . 4 ⊢ ((*Q‘𝑥) ·Q 𝑥) = (𝑥 ·Q (*Q‘𝑥)) | |
| 5 | recidnq 11005 | . . . 4 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
| 6 | 4, 5 | eqtrid 2789 | . . 3 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 𝑥) = 1Q) | 
| 7 | recclnq 11006 | . . . 4 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
| 8 | recmulnq 11004 | . . . 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 257 | . 2 ⊢ (𝑥 ∈ Q → (*Q‘(*Q‘𝑥)) = 𝑥) | 
| 11 | 3, 10 | vtoclga 3577 | 1 ⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ‘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: reclem2pr 11088 | 
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