Theorem recidpipr 7024

Description: Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)

Assertion

Ref	Expression
recidpipr	⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = 1P)

Distinct variable group:   𝑁,𝑙,𝑢

Proof of Theorem recidpipr

Step	Hyp	Ref	Expression
1		nnnq 6612	. . 3 ⊢ (𝑁 ∈ N → [⟨𝑁, 1𝑜⟩] ~Q ∈ Q)
2		recclnq 6582	. . . 4 ⊢ ([⟨𝑁, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) ∈ Q)
3	1, 2	syl 14	. . 3 ⊢ (𝑁 ∈ N → (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) ∈ Q)
4		mulnqpr 6767	. . 3 ⊢ (([⟨𝑁, 1𝑜⟩] ~Q ∈ Q ∧ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
5	1, 3, 4	syl2anc 403	. 2 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
6		recidnq 6583	. . . . . . 7 ⊢ ([⟨𝑁, 1𝑜⟩] ~Q ∈ Q → ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) = 1Q)
7	1, 6	syl 14	. . . . . 6 ⊢ (𝑁 ∈ N → ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) = 1Q)
8	7	breq2d 3797	. . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) ↔ 𝑙 <Q 1Q))
9	8	abbidv 2196	. . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q})
10	7	breq1d 3795	. . . . 5 ⊢ (𝑁 ∈ N → (([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢))
11	10	abbidv 2196	. . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
12	9, 11	opeq12d 3578	. . 3 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩)
13		df-i1p 6657	. . 3 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
14	12, 13	syl6eqr 2131	. 2 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = 1P)
15	5, 14	eqtr3d 2115	1 ⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1𝑜⟩] ~Q <Q 𝑢}⟩ ·P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) = 1P)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  {cab 2067  ⟨cop 3401   class class class wbr 3785  ‘cfv 4922  (class class class)co 5532  1_𝑜c1o 6017  [cec 6127  Ncnpi 6462   ~_Q ceq 6469  Qcnq 6470  1_Qc1q 6471   ·_Q cmq 6473  *_Qcrq 6474   <_Q cltq 6475  1_Pc1p 6482   ·_P cmp 6484

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-imp 6659

This theorem is referenced by:  recidpirq  7026