Theorem caucvgprprlemloccalc 6782

Description: Lemma for caucvgprpr 6810. Rearranging some expressions for caucvgprprlemloc 6801. (Contributed by Jim Kingdon, 8-Feb-2021.)

Hypotheses

Ref	Expression
caucvgprprlemloccalc.st	⊢ (𝜑 → 𝑆 <Q 𝑇)
caucvgprprlemloccalc.y	⊢ (𝜑 → 𝑌 ∈ Q)
caucvgprprlemloccalc.syt	⊢ (𝜑 → (𝑆 +Q 𝑌) = 𝑇)
caucvgprprlemloccalc.x	⊢ (𝜑 → 𝑋 ∈ Q)
caucvgprprlemloccalc.xxy	⊢ (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)
caucvgprprlemloccalc.m	⊢ (𝜑 → 𝑀 ∈ N)
caucvgprprlemloccalc.mx	⊢ (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)

Assertion

Ref	Expression
caucvgprprlemloccalc	⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝑇}, {𝑢 ∣ 𝑇 <Q 𝑢}⟩)

Distinct variable groups:   𝑀,𝑙,𝑢   𝑆,𝑙,𝑢   𝑇,𝑙,𝑢

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝑋(𝑢,𝑙)   𝑌(𝑢,𝑙)

Proof of Theorem caucvgprprlemloccalc

Step	Hyp	Ref	Expression
1		caucvgprprlemloccalc.st	. . . . . 6 ⊢ (𝜑 → 𝑆 <Q 𝑇)
2		ltrelnq 6463	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4392	. . . . . 6 ⊢ (𝑆 <Q 𝑇 → (𝑆 ∈ Q ∧ 𝑇 ∈ Q))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ Q ∧ 𝑇 ∈ Q))
5	4	simpld 105	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Q)
6		caucvgprprlemloccalc.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ N)
7		nnnq 6520	. . . . 5 ⊢ (𝑀 ∈ N → [⟨𝑀, 1𝑜⟩] ~Q ∈ Q)
8		recclnq 6490	. . . . 5 ⊢ ([⟨𝑀, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q)
9	6, 7, 8	3syl 17	. . . 4 ⊢ (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q)
10		addclnq 6473	. . . 4 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q)
11	5, 9, 10	syl2anc 391	. . 3 ⊢ (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q)
12		addnqpr 6659	. . 3 ⊢ (((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → ⟨{𝑙 ∣ 𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
13	11, 9, 12	syl2anc 391	. 2 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ = (⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))
14		addassnqg 6480	. . . . 5 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) = (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))))
15	5, 9, 9, 14	syl3anc 1135	. . . 4 ⊢ (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) = (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))))
16		caucvgprprlemloccalc.mx	. . . . . . . 8 ⊢ (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋)
17		caucvgprprlemloccalc.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ Q)
18		lt2addnq 6502	. . . . . . . . 9 ⊢ ((((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q ∧ 𝑋 ∈ Q) ∧ ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q ∧ 𝑋 ∈ Q)) → (((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋 ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋) → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋)))
19	9, 17, 9, 17, 18	syl22anc 1136	. . . . . . . 8 ⊢ (𝜑 → (((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋 ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑋) → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋)))
20	16, 16, 19	mp2and 409	. . . . . . 7 ⊢ (𝜑 → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋))
21		caucvgprprlemloccalc.xxy	. . . . . . 7 ⊢ (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)
22		ltsonq 6496	. . . . . . . 8 ⊢ <Q Or Q
23	22, 2	sotri 4720	. . . . . . 7 ⊢ ((((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q (𝑋 +Q 𝑋) ∧ (𝑋 +Q 𝑋) <Q 𝑌) → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌)
24	20, 21, 23	syl2anc 391	. . . . . 6 ⊢ (𝜑 → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌)
25		ltanqi 6500	. . . . . 6 ⊢ ((((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌 ∧ 𝑆 ∈ Q) → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q (𝑆 +Q 𝑌))
26	24, 5, 25	syl2anc 391	. . . . 5 ⊢ (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q (𝑆 +Q 𝑌))
27		caucvgprprlemloccalc.syt	. . . . 5 ⊢ (𝜑 → (𝑆 +Q 𝑌) = 𝑇)
28	26, 27	breqtrd 3788	. . . 4 ⊢ (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q 𝑇)
29	15, 28	eqbrtrd 3784	. . 3 ⊢ (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑇)
30		ltnqpri 6692	. . 3 ⊢ (((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑇 → ⟨{𝑙 ∣ 𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑇}, {𝑢 ∣ 𝑇 <Q 𝑢}⟩)
31	29, 30	syl 14	. 2 ⊢ (𝜑 → ⟨{𝑙 ∣ 𝑙 <Q ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝑇}, {𝑢 ∣ 𝑇 <Q 𝑢}⟩)
32	13, 31	eqbrtrrd 3786	1 ⊢ (𝜑 → (⟨{𝑙 ∣ 𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙 ∣ 𝑙 <Q 𝑇}, {𝑢 ∣ 𝑇 <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 97   = 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   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383   +_P cpp 6391  <_P cltp 6393

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-iplp 6566  df-iltp 6568

This theorem is referenced by:  caucvgprprlemloc  6801