Theorem caucvgprprlemnkj 7754

Description: Lemma for caucvgprpr 7774. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	⊢ (𝜑 → 𝐹:N⟶P)
caucvgprpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprprlemnkj.k	⊢ (𝜑 → 𝐾 ∈ N)
caucvgprprlemnkj.j	⊢ (𝜑 → 𝐽 ∈ N)
caucvgprprlemnkj.s	⊢ (𝜑 → 𝑆 ∈ Q)

Assertion

Ref	Expression
caucvgprprlemnkj	⊢ (𝜑 → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))

Distinct variable groups:   𝑘,𝐹,𝑛   𝐽,𝑝,𝑞   𝑢,𝐽   𝐽,𝑙   𝐾,𝑝,𝑞   𝐾,𝑙   𝑢,𝐾   𝑆,𝑝,𝑞   𝑢,𝑛   𝑛,𝑙,𝑘   𝑢,𝑘   𝑢,𝑞   𝑝,𝑙

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑆(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑞,𝑝,𝑙)   𝐽(𝑘,𝑛)   𝐾(𝑘,𝑛)

Proof of Theorem caucvgprprlemnkj

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . 3 ⊢ (𝜑 → 𝐹:N⟶P)
2		caucvgprpr.cau	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
3		caucvgprprlemnkj.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ N)
4		caucvgprprlemnkj.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ N)
5		caucvgprprlemnkj.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ Q)
6	1, 2, 3, 4, 5	caucvgprprlemnkltj 7751	. 2 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
7	1, 2, 3, 4, 5	caucvgprprlemnkeqj 7752	. 2 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
8	1, 2, 3, 4, 5	caucvgprprlemnjltk 7753	. 2 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
9		pitri3or 7384	. . 3 ⊢ ((𝐾 ∈ N ∧ 𝐽 ∈ N) → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾))
10	3, 4, 9	syl2anc 411	. 2 ⊢ (𝜑 → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾))
11	6, 7, 8, 10	mpjao3dan 1318	1 ⊢ (𝜑 → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ w3o 979   = wceq 1364   ∈ wcel 2164  {cab 2179  ∀wral 2472  ⟨cop 3622   class class class wbr 4030  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  1_oc1o 6464  [cec 6587  Ncnpi 7334   <_N clti 7337   ~_Q ceq 7341  Qcnq 7342   +_Q cplq 7344  *_Qcrq 7346   <_Q cltq 7347  Pcnp 7353   +_P cpp 7355  <_P cltp 7357

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iplp 7530  df-iltp 7532

This theorem is referenced by:  caucvgprprlemdisj  7764