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Theorem caucvgprprlemloccalc 7460
Description: Lemma for caucvgprpr 7488. Rearranging some expressions for caucvgprprlemloc 7479. (Contributed by Jim Kingdon, 8-Feb-2021.)
Hypotheses
Ref Expression
caucvgprprlemloccalc.st (𝜑𝑆 <Q 𝑇)
caucvgprprlemloccalc.y (𝜑𝑌Q)
caucvgprprlemloccalc.syt (𝜑 → (𝑆 +Q 𝑌) = 𝑇)
caucvgprprlemloccalc.x (𝜑𝑋Q)
caucvgprprlemloccalc.xxy (𝜑 → (𝑋 +Q 𝑋) <Q 𝑌)
caucvgprprlemloccalc.m (𝜑𝑀N)
caucvgprprlemloccalc.mx (𝜑 → (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑋)
Assertion
Ref Expression
caucvgprprlemloccalc (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <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
StepHypRef Expression
1 caucvgprprlemloccalc.st . . . . . 6 (𝜑𝑆 <Q 𝑇)
2 ltrelnq 7141 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4561 . . . . . 6 (𝑆 <Q 𝑇 → (𝑆Q𝑇Q))
41, 3syl 14 . . . . 5 (𝜑 → (𝑆Q𝑇Q))
54simpld 111 . . . 4 (𝜑𝑆Q)
6 caucvgprprlemloccalc.m . . . . 5 (𝜑𝑀N)
7 nnnq 7198 . . . . 5 (𝑀N → [⟨𝑀, 1o⟩] ~QQ)
8 recclnq 7168 . . . . 5 ([⟨𝑀, 1o⟩] ~QQ → (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q)
96, 7, 83syl 17 . . . 4 (𝜑 → (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q)
10 addclnq 7151 . . . 4 ((𝑆Q ∧ (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
115, 9, 10syl2anc 408 . . 3 (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
12 addnqpr 7337 . . 3 (((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q) → ⟨{𝑙𝑙 <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}⟩ = (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑢}⟩))
1311, 9, 12syl2anc 408 . 2 (𝜑 → ⟨{𝑙𝑙 <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}⟩ = (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑢}⟩))
14 addassnqg 7158 . . . . 5 ((𝑆Q ∧ (*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 ))))
155, 9, 9, 14syl3anc 1201 . . . 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 (𝜑𝑋Q)
18 lt2addnq 7180 . . . . . . . . 9 ((((*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q𝑋Q) ∧ ((*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q𝑋Q)) → (((*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)))
199, 17, 9, 17, 18syl22anc 1202 . . . . . . . 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)))
2016, 16, 19mp2and 429 . . . . . . 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 7174 . . . . . . . 8 <Q Or Q
2322, 2sotri 4904 . . . . . . 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)
2420, 21, 23syl2anc 408 . . . . . 6 (𝜑 → ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑌)
25 ltanqi 7178 . . . . . 6 ((((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑌𝑆Q) → (𝑆 +Q ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))) <Q (𝑆 +Q 𝑌))
2624, 5, 25syl2anc 408 . . . . 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)
2826, 27breqtrd 3924 . . . 4 (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))) <Q 𝑇)
2915, 28eqbrtrd 3920 . . 3 (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑇)
30 ltnqpri 7370 . . 3 (((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑇 → ⟨{𝑙𝑙 <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 ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
3129, 30syl 14 . 2 (𝜑 → ⟨{𝑙𝑙 <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 ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
3213, 31eqbrtrrd 3922 1 (𝜑 → (⟨{𝑙𝑙 <Q (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))}, {𝑢 ∣ (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑢}⟩ +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑀, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑀, 1o⟩] ~Q ) <Q 𝑢}⟩)<P ⟨{𝑙𝑙 <Q 𝑇}, {𝑢𝑇 <Q 𝑢}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  {cab 2103  cop 3500   class class class wbr 3899  cfv 5093  (class class class)co 5742  1oc1o 6274  [cec 6395  Ncnpi 7048   ~Q ceq 7055  Qcnq 7056   +Q cplq 7058  *Qcrq 7060   <Q cltq 7061   +P cpp 7069  <P cltp 7071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  caucvgprprlemloc  7479
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