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Mirrors > Home > ILE Home > Th. List > caucvgprprlemnkj | GIF version |
Description: Lemma for caucvgprpr 7544. 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 7521 | . 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 7522 | . 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 7523 | . 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 7154 | . . 3 ⊢ ((𝐾 ∈ N ∧ 𝐽 ∈ N) → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾)) | |
10 | 3, 4, 9 | syl2anc 409 | . 2 ⊢ (𝜑 → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾)) |
11 | 6, 7, 8, 10 | mpjao3dan 1286 | 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 962 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 〈cop 3535 class class class wbr 3937 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 1oc1o 6314 [cec 6435 Ncnpi 7104 <N clti 7107 ~Q ceq 7111 Qcnq 7112 +Q cplq 7114 *Qcrq 7116 <Q cltq 7117 Pcnp 7123 +P cpp 7125 <P cltp 7127 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-eprel 4219 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-2o 6322 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-lti 7139 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184 df-ltnqqs 7185 df-enq0 7256 df-nq0 7257 df-0nq0 7258 df-plq0 7259 df-mq0 7260 df-inp 7298 df-iplp 7300 df-iltp 7302 |
This theorem is referenced by: caucvgprprlemdisj 7534 |
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