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Mirrors > Home > ILE Home > Th. List > caucvgprprlemnkj | GIF version |
Description: Lemma for caucvgprpr 7513. 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 7490 | . 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 7491 | . 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 7492 | . 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 7123 | . . 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 1285 | 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 961 = wceq 1331 ∈ wcel 1480 {cab 2123 ∀wral 2414 〈cop 3525 class class class wbr 3924 ⟶wf 5114 ‘cfv 5118 (class class class)co 5767 1oc1o 6299 [cec 6420 Ncnpi 7073 <N clti 7076 ~Q ceq 7080 Qcnq 7081 +Q cplq 7083 *Qcrq 7085 <Q cltq 7086 Pcnp 7092 +P cpp 7094 <P cltp 7096 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-eprel 4206 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-1o 6306 df-2o 6307 df-oadd 6310 df-omul 6311 df-er 6422 df-ec 6424 df-qs 6428 df-ni 7105 df-pli 7106 df-mi 7107 df-lti 7108 df-plpq 7145 df-mpq 7146 df-enq 7148 df-nqqs 7149 df-plqqs 7150 df-mqqs 7151 df-1nqqs 7152 df-rq 7153 df-ltnqqs 7154 df-enq0 7225 df-nq0 7226 df-0nq0 7227 df-plq0 7228 df-mq0 7229 df-inp 7267 df-iplp 7269 df-iltp 7271 |
This theorem is referenced by: caucvgprprlemdisj 7503 |
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