Theorem recidpipr 8011

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 7577	. . 3 ⊢ (𝑁 ∈ N → [⟨𝑁, 1o⟩] ~Q ∈ Q)
2		recclnq 7547	. . . 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 7732	. . 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 7548	. . . . . . 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 4074	. . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) ↔ 𝑙 <Q 1Q))
9	8	abbidv 2327	. . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q})
10	7	breq1d 4072	. . . . 5 ⊢ (𝑁 ∈ N → (([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢))
11	10	abbidv 2327	. . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
12	9, 11	opeq12d 3844	. . 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 7622	. . 3 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
14	12, 13	eqtr4di 2260	. 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 2244	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 1375   ∈ wcel 2180  {cab 2195  ⟨cop 3649   class class class wbr 4062  ‘cfv 5294  (class class class)co 5974  1_oc1o 6525  [cec 6648  Ncnpi 7427   ~_Q ceq 7434  Qcnq 7435  1_Qc1q 7436   ·_Q cmq 7438  *_Qcrq 7439   <_Q cltq 7440  1_Pc1p 7447   ·_P cmp 7449

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-eprel 4357  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-1o 6532  df-2o 6533  df-oadd 6536  df-omul 6537  df-er 6650  df-ec 6652  df-qs 6656  df-ni 7459  df-pli 7460  df-mi 7461  df-lti 7462  df-plpq 7499  df-mpq 7500  df-enq 7502  df-nqqs 7503  df-plqqs 7504  df-mqqs 7505  df-1nqqs 7506  df-rq 7507  df-ltnqqs 7508  df-enq0 7579  df-nq0 7580  df-0nq0 7581  df-plq0 7582  df-mq0 7583  df-inp 7621  df-i1p 7622  df-imp 7624

This theorem is referenced by:  recidpirq  8013