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Theorem caucvgprprlemloccalc 6840
Description: Lemma for caucvgprpr 6868. Rearranging some expressions for caucvgprprlemloc 6859. (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 6521 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4420 . . . . . 6 (𝑆 <Q 𝑇 → (𝑆Q𝑇Q))
41, 3syl 14 . . . . 5 (𝜑 → (𝑆Q𝑇Q))
54simpld 109 . . . 4 (𝜑𝑆Q)
6 caucvgprprlemloccalc.m . . . . 5 (𝜑𝑀N)
7 nnnq 6578 . . . . 5 (𝑀N → [⟨𝑀, 1𝑜⟩] ~QQ)
8 recclnq 6548 . . . . 5 ([⟨𝑀, 1𝑜⟩] ~QQ → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q)
96, 7, 83syl 17 . . . 4 (𝜑 → (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q)
10 addclnq 6531 . . . 4 ((𝑆Q ∧ (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q)
115, 9, 10syl2anc 397 . . 3 (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) ∈ Q)
12 addnqpr 6717 . . 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 397 . 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 6538 . . . . 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 1146 . . . 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 6560 . . . . . . . . 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 1147 . . . . . . . 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 417 . . . . . . 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 6554 . . . . . . . 8 <Q Or Q
2322, 2sotri 4748 . . . . . . 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 397 . . . . . 6 (𝜑 → ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑌)
25 ltanqi 6558 . . . . . 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 397 . . . . 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 3816 . . . 4 (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1𝑜⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q ))) <Q 𝑇)
2915, 28eqbrtrd 3812 . . 3 (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1𝑜⟩] ~Q )) <Q 𝑇)
30 ltnqpri 6750 . . 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 3814 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 101   = wceq 1259  wcel 1409  {cab 2042  cop 3406   class class class wbr 3792  cfv 4930  (class class class)co 5540  1𝑜c1o 6025  [cec 6135  Ncnpi 6428   ~Q ceq 6435  Qcnq 6436   +Q cplq 6438  *Qcrq 6440   <Q cltq 6441   +P cpp 6449  <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  caucvgprprlemloc  6859
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