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Mirrors > Home > ILE Home > Th. List > caucvgprprlemnkj | GIF version |
Description: Lemma for caucvgprpr 7488. Part of disjointness. (Contributed by Jim Kingdon, 20-Jan-2021.) |
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) |
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}〉)) |
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 7465 | . 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 7466 | . 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 7467 | . 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 7098 | . . 3 ⊢ ((𝐾 ∈ N ∧ 𝐽 ∈ N) → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾)) | |
10 | 3, 4, 9 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾)) |
11 | 6, 7, 8, 10 | mpjao3dan 1270 | 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}〉)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ w3o 946 = wceq 1316 ∈ wcel 1465 {cab 2103 ∀wral 2393 〈cop 3500 class class class wbr 3899 ⟶wf 5089 ‘cfv 5093 (class class class)co 5742 1oc1o 6274 [cec 6395 Ncnpi 7048 <N clti 7051 ~Q ceq 7055 Qcnq 7056 +Q cplq 7058 *Qcrq 7060 <Q cltq 7061 Pcnp 7067 +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: caucvgprprlemdisj 7478 |
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