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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 6912 | . . 3 ⊢ (𝑥 = 𝐴 → (*Q‘(*Q‘𝑥)) = (*Q‘(*Q‘𝐴))) | |
2 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
3 | 1, 2 | eqeq12d 2751 | . 2 ⊢ (𝑥 = 𝐴 → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ (*Q‘(*Q‘𝐴)) = 𝐴)) |
4 | mulcomnq 10991 | . . . 4 ⊢ ((*Q‘𝑥) ·Q 𝑥) = (𝑥 ·Q (*Q‘𝑥)) | |
5 | recidnq 11003 | . . . 4 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
6 | 4, 5 | eqtrid 2787 | . . 3 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 𝑥) = 1Q) |
7 | recclnq 11004 | . . . 4 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
8 | recmulnq 11002 | . . . 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 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Qcnq 10890 1Qc1q 10891 ·Q cmq 10894 *Qcrq 10895 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ni 10910 df-mi 10912 df-lti 10913 df-mpq 10947 df-enq 10949 df-nq 10950 df-erq 10951 df-mq 10953 df-1nq 10954 df-rq 10955 |
This theorem is referenced by: reclem2pr 11086 |
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