Theorem recidpipr 8069

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 7635	. . 3 ⊢ (𝑁 ∈ N → [⟨𝑁, 1o⟩] ~Q ∈ Q)
2		recclnq 7605	. . . 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 7790	. . 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 411	. 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 7606	. . . . . . 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 4098	. . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) ↔ 𝑙 <Q 1Q))
9	8	abbidv 2347	. . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q})
10	7	breq1d 4096	. . . . 5 ⊢ (𝑁 ∈ N → (([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢))
11	10	abbidv 2347	. . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
12	9, 11	opeq12d 3868	. . 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 7680	. . 3 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
14	12, 13	eqtr4di 2280	. 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 2264	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 1395   ∈ wcel 2200  {cab 2215  ⟨cop 3670   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  1_oc1o 6570  [cec 6695  Ncnpi 7485   ~_Q ceq 7492  Qcnq 7493  1_Qc1q 7494   ·_Q cmq 7496  *_Qcrq 7497   <_Q cltq 7498  1_Pc1p 7505   ·_P cmp 7507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-2o 6578  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-enq0 7637  df-nq0 7638  df-0nq0 7639  df-plq0 7640  df-mq0 7641  df-inp 7679  df-i1p 7680  df-imp 7682

This theorem is referenced by:  recidpirq  8071