Theorem recidpipr 6932

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 6520	. . 3 ⊢ (𝑁 ∈ N → [⟨𝑁, 1𝑜⟩] ~Q ∈ Q)
2		recclnq 6490	. . . 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 6675	. . 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 391	. 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 6491	. . . . . . 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 3776	. . . . 5 ⊢ (𝑁 ∈ N → (𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) ↔ 𝑙 <Q 1Q))
9	8	abbidv 2155	. . . 4 ⊢ (𝑁 ∈ N → {𝑙 ∣ 𝑙 <Q ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q ))} = {𝑙 ∣ 𝑙 <Q 1Q})
10	7	breq1d 3774	. . . . 5 ⊢ (𝑁 ∈ N → (([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢 ↔ 1Q <Q 𝑢))
11	10	abbidv 2155	. . . 4 ⊢ (𝑁 ∈ N → {𝑢 ∣ ([⟨𝑁, 1𝑜⟩] ~Q ·Q (*Q‘[⟨𝑁, 1𝑜⟩] ~Q )) <Q 𝑢} = {𝑢 ∣ 1Q <Q 𝑢})
12	9, 11	opeq12d 3557	. . 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 6565	. . 3 ⊢ 1P = ⟨{𝑙 ∣ 𝑙 <Q 1Q}, {𝑢 ∣ 1Q <Q 𝑢}⟩
14	12, 13	syl6eqr 2090	. 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 2074	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 1243   ∈ wcel 1393  {cab 2026  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~_Q ceq 6377  Qcnq 6378  1_Qc1q 6379   ·_Q cmq 6381  *_Qcrq 6382   <_Q cltq 6383  1_Pc1p 6390   ·_P cmp 6392

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-i1p 6565  df-imp 6567

This theorem is referenced by:  recidpirq  6934