Theorem recidpipr 7918

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 7484	. . 3 ⊢ (𝑁 ∈ N → [⟨𝑁, 1o⟩] ~Q ∈ Q)
2		recclnq 7454	. . . 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 7639	. . 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 7455	. . . . . . 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 4042	. . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) ↔ 𝑙 <Q 1Q))
9	8	abbidv 2311	. . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q})
10	7	breq1d 4040	. . . . 5 ⊢ (𝑁 ∈ N → (([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢))
11	10	abbidv 2311	. . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([⟨𝑁, 1o⟩] ~Q ·Q (*Q‘[⟨𝑁, 1o⟩] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
12	9, 11	opeq12d 3813	. . 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 7529	. . 3 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
14	12, 13	eqtr4di 2244	. 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 2228	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 1364   ∈ wcel 2164  {cab 2179  ⟨cop 3622   class class class wbr 4030  ‘cfv 5255  (class class class)co 5919  1_oc1o 6464  [cec 6587  Ncnpi 7334   ~_Q ceq 7341  Qcnq 7342  1_Qc1q 7343   ·_Q cmq 7345  *_Qcrq 7346   <_Q cltq 7347  1_Pc1p 7354   ·_P cmp 7356

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-i1p 7529  df-imp 7531

This theorem is referenced by:  recidpirq  7920