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Theorem caucvgprprlemloccalc 7682
Description: Lemma for caucvgprpr 7710. Rearranging some expressions for caucvgprprlemloc 7701. (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 7363 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4678 . . . . . 6 (𝑆 <Q 𝑇 → (𝑆Q𝑇Q))
41, 3syl 14 . . . . 5 (𝜑 → (𝑆Q𝑇Q))
54simpld 112 . . . 4 (𝜑𝑆Q)
6 caucvgprprlemloccalc.m . . . . 5 (𝜑𝑀N)
7 nnnq 7420 . . . . 5 (𝑀N → [⟨𝑀, 1o⟩] ~QQ)
8 recclnq 7390 . . . . 5 ([⟨𝑀, 1o⟩] ~QQ → (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q)
96, 7, 83syl 17 . . . 4 (𝜑 → (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q)
10 addclnq 7373 . . . 4 ((𝑆Q ∧ (*Q‘[⟨𝑀, 1o⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
115, 9, 10syl2anc 411 . . 3 (𝜑 → (𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) ∈ Q)
12 addnqpr 7559 . . 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 411 . 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 7380 . . . . 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 1238 . . . 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 7402 . . . . . . . . 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 1239 . . . . . . . 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 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 7396 . . . . . . . 8 <Q Or Q
2322, 2sotri 5024 . . . . . . 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 411 . . . . . 6 (𝜑 → ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑌)
25 ltanqi 7400 . . . . . 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 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)
2826, 27breqtrd 4029 . . . 4 (𝜑 → (𝑆 +Q ((*Q‘[⟨𝑀, 1o⟩] ~Q ) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q ))) <Q 𝑇)
2915, 28eqbrtrd 4025 . . 3 (𝜑 → ((𝑆 +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) +Q (*Q‘[⟨𝑀, 1o⟩] ~Q )) <Q 𝑇)
30 ltnqpri 7592 . . 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 4027 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 104   = wceq 1353  wcel 2148  {cab 2163  cop 3595   class class class wbr 4003  cfv 5216  (class class class)co 5874  1oc1o 6409  [cec 6532  Ncnpi 7270   ~Q ceq 7277  Qcnq 7278   +Q cplq 7280  *Qcrq 7282   <Q cltq 7283   +P cpp 7291  <P cltp 7293
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468
This theorem is referenced by:  caucvgprprlemloc  7701
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