Theorem caucvgprprlemloccalc 7393

Description: Lemma for caucvgprpr 7421. Rearranging some expressions for caucvgprprlemloc 7412. (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 7074	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4529	. . . . . 6 ⊢ (𝑆 <Q 𝑇 → (𝑆 ∈ Q ∧ 𝑇 ∈ Q))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ Q ∧ 𝑇 ∈ Q))
5	4	simpld 111	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Q)
6		caucvgprprlemloccalc.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ N)
7		nnnq 7131	. . . . 5 ⊢ (𝑀 ∈ N → [⟨𝑀, 1o⟩] ~Q ∈ Q)
8		recclnq 7101	. . . . 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 7084	. . . 4 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
11	5, 9, 10	syl2anc 406	. . 3 ⊢ (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
12		addnqpr 7270	. . 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 406	. 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 7091	. . . . 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 1184	. . . 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 7113	. . . . . . . . 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 1185	. . . . . . . 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 427	. . . . . . 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 7107	. . . . . . . 8 ⊢ <Q Or Q
23	22, 2	sotri 4870	. . . . . . 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 406	. . . . . 6 ⊢ (𝜑 → ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑌)
25		ltanqi 7111	. . . . . 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 406	. . . . 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 3899	. . . 4 ⊢ (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))) <Q 𝑇)
29	15, 28	eqbrtrd 3895	. . 3 ⊢ (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑇)
30		ltnqpri 7303	. . 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 3897	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 103   = wceq 1299   ∈ wcel 1448  {cab 2086  ⟨cop 3477   class class class wbr 3875  ‘cfv 5059  (class class class)co 5706  1_oc1o 6236  [cec 6357  Ncnpi 6981   ~_Q ceq 6988  Qcnq 6989   +_Q cplq 6991  *_Qcrq 6993   <_Q cltq 6994   +_P cpp 7002  <_P cltp 7004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-iltp 7179

This theorem is referenced by:  caucvgprprlemloc  7412