Theorem caucvgprprlemloccalc 7882

Description: Lemma for caucvgprpr 7910. Rearranging some expressions for caucvgprprlemloc 7901. (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 7563	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4771	. . . . . 6 ⊢ (𝑆 <Q 𝑇 → (𝑆 ∈ Q ∧ 𝑇 ∈ Q))
4	1, 3	syl 14	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ Q ∧ 𝑇 ∈ Q))
5	4	simpld 112	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Q)
6		caucvgprprlemloccalc.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ N)
7		nnnq 7620	. . . . 5 ⊢ (𝑀 ∈ N → [⟨𝑀, 1o⟩] ~Q ∈ Q)
8		recclnq 7590	. . . . 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 7573	. . . 4 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
11	5, 9, 10	syl2anc 411	. . 3 ⊢ (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
12		addnqpr 7759	. . 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 411	. 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 7580	. . . . 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 1271	. . . 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 7602	. . . . . . . . 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 1272	. . . . . . . 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 433	. . . . . . 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 7596	. . . . . . . 8 ⊢ <Q Or Q
23	22, 2	sotri 5124	. . . . . . 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 411	. . . . . 6 ⊢ (𝜑 → ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑌)
25		ltanqi 7600	. . . . . 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 411	. . . . 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 4109	. . . 4 ⊢ (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))) <Q 𝑇)
29	15, 28	eqbrtrd 4105	. . 3 ⊢ (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑇)
30		ltnqpri 7792	. . 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 4107	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 104   = wceq 1395   ∈ wcel 2200  {cab 2215  ⟨cop 3669   class class class wbr 4083  ‘cfv 5318  (class class class)co 6007  1_oc1o 6561  [cec 6686  Ncnpi 7470   ~_Q ceq 7477  Qcnq 7478   +_Q cplq 7480  *_Qcrq 7482   <_Q cltq 7483   +_P cpp 7491  <_P cltp 7493

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-irdg 6522  df-1o 6568  df-2o 6569  df-oadd 6572  df-omul 6573  df-er 6688  df-ec 6690  df-qs 6694  df-ni 7502  df-pli 7503  df-mi 7504  df-lti 7505  df-plpq 7542  df-mpq 7543  df-enq 7545  df-nqqs 7546  df-plqqs 7547  df-mqqs 7548  df-1nqqs 7549  df-rq 7550  df-ltnqqs 7551  df-enq0 7622  df-nq0 7623  df-0nq0 7624  df-plq0 7625  df-mq0 7626  df-inp 7664  df-iplp 7666  df-iltp 7668

This theorem is referenced by:  caucvgprprlemloc  7901