Theorem caucvgprprlemloccalc 8057

Description: Lemma for caucvgprpr 8085. Rearranging some expressions for caucvgprprlemloc 8076. (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 7738	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
3	2	brel 4829	. . . . . 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 7795	. . . . 5 ⊢ (𝑀 ∈ N → [⟨𝑀, 1o⟩] ~Q ∈ Q)
8		recclnq 7765	. . . . 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 7748	. . . 4 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
11	5, 9, 10	syl2anc 415	. . 3 ⊢ (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
12		addnqpr 7934	. . 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 415	. 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 7755	. . . . 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 1278	. . . 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 7777	. . . . . . . . 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 1279	. . . . . . . 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 437	. . . . . . 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 7771	. . . . . . . 8 ⊢ <Q Or Q
23	22, 2	sotri 5186	. . . . . . 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 415	. . . . . 6 ⊢ (𝜑 → ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑌)
25		ltanqi 7775	. . . . . 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 415	. . . . 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 4157	. . . 4 ⊢ (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))) <Q 𝑇)
29	15, 28	eqbrtrd 4153	. . 3 ⊢ (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑇)
30		ltnqpri 7967	. . 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 4155	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}⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ⟨cop 3712   class class class wbr 4131  ‘cfv 5380  (class class class)co 6088  1_oc1o 6684  [cec 6809  Ncnpi 7645   ~_Q ceq 7652  Qcnq 7653   +_Q cplq 7655  *_Qcrq 7657   <_Q cltq 7658   +_P cpp 7666  <_P cltp 7668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843

This theorem is used by:  caucvgprprlemloc  8076