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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 | fveq2 6191 | . . . 4 ⊢ (𝑥 = 𝐴 → (*Q‘𝑥) = (*Q‘𝐴)) | |
2 | 1 | fveq2d 6195 | . . 3 ⊢ (𝑥 = 𝐴 → (*Q‘(*Q‘𝑥)) = (*Q‘(*Q‘𝐴))) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2637 | . 2 ⊢ (𝑥 = 𝐴 → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ (*Q‘(*Q‘𝐴)) = 𝐴)) |
5 | mulcomnq 9775 | . . . 4 ⊢ ((*Q‘𝑥) ·Q 𝑥) = (𝑥 ·Q (*Q‘𝑥)) | |
6 | recidnq 9787 | . . . 4 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q) | |
7 | 5, 6 | syl5eq 2668 | . . 3 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 𝑥) = 1Q) |
8 | recclnq 9788 | . . . 4 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q) | |
9 | recmulnq 9786 | . . . 4 ⊢ ((*Q‘𝑥) ∈ Q → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ ((*Q‘𝑥) ·Q 𝑥) = 1Q)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑥 ∈ Q → ((*Q‘(*Q‘𝑥)) = 𝑥 ↔ ((*Q‘𝑥) ·Q 𝑥) = 1Q)) |
11 | 7, 10 | mpbird 247 | . 2 ⊢ (𝑥 ∈ Q → (*Q‘(*Q‘𝑥)) = 𝑥) |
12 | 4, 11 | vtoclga 3272 | 1 ⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Qcnq 9674 1Qc1q 9675 ·Q cmq 9678 *Qcrq 9679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-mi 9696 df-lti 9697 df-mpq 9731 df-enq 9733 df-nq 9734 df-erq 9735 df-mq 9737 df-1nq 9738 df-rq 9739 |
This theorem is referenced by: reclem2pr 9870 |
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