Theorem recrecnq 10389

Description: Reciprocal of reciprocal of positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 29-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
recrecnq	⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴)

Proof of Theorem recrecnq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2fveq3 6675	. . 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 10375	. . . 4 ⊢ ((*Q‘𝑥) ·Q 𝑥) = (𝑥 ·Q (*Q‘𝑥))
5		recidnq 10387	. . . 4 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
6	4, 5	syl5eq 2868	. . 3 ⊢ (𝑥 ∈ Q → ((*Q‘𝑥) ·Q 𝑥) = 1Q)
7		recclnq 10388	. . . 4 ⊢ (𝑥 ∈ Q → (*Q‘𝑥) ∈ Q)
8		recmulnq 10386	. . . 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 3574	1 ⊢ (𝐴 ∈ Q → (*Q‘(*Q‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537   ∈ wcel 2114  ‘cfv 6355  (class class class)co 7156  Qcnq 10274  1_Qc1q 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:  reclem2pr  10470