Theorem recidpipr 8076

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 7642	. . 3 ⊢ (𝑁 ∈ N → [⟨𝑁, 1o⟩] ~Q ∈ Q)
2		recclnq 7612	. . . 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 7797	. . 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 7613	. . . . . . 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 4100	. . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) ↔ 𝑙 <Q 1Q))
9	8	abbidv 2349	. . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q})
10	7	breq1d 4098	. . . . 5 ⊢ (𝑁 ∈ N → (([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢))
11	10	abbidv 2349	. . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
12	9, 11	opeq12d 3870	. . 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 7687	. . 3 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
14	12, 13	eqtr4di 2282	. 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 2266	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 1397   ∈ wcel 2202  {cab 2217  ⟨cop 3672   class class class wbr 4088  ‘cfv 5326  (class class class)co 6018  1_oc1o 6575  [cec 6700  Ncnpi 7492   ~_Q ceq 7499  Qcnq 7500  1_Qc1q 7501   ·_Q cmq 7503  *_Qcrq 7504   <_Q cltq 7505  1_Pc1p 7512   ·_P cmp 7514

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-2o 6583  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-enq0 7644  df-nq0 7645  df-0nq0 7646  df-plq0 7647  df-mq0 7648  df-inp 7686  df-i1p 7687  df-imp 7689

This theorem is referenced by:  recidpirq  8078