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Mirrors > Home > ILE Home > Th. List > prarloclem4 | GIF version |
Description: A slight rearrangement of prarloclem3 7305. Lemma for prarloc 7311. (Contributed by Jim Kingdon, 4-Nov-2019.) |
Ref | Expression |
---|---|
prarloclem4 | ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ 𝑃 ∈ Q) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prarloclem3 7305 | . . . . 5 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑥 ∈ ω ∧ 𝑃 ∈ Q) ∧ ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈)) | |
2 | 1 | 3expia 1183 | . . . 4 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑥 ∈ ω ∧ 𝑃 ∈ Q)) → (∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
3 | 2 | ancom2s 555 | . . 3 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ (𝑃 ∈ Q ∧ 𝑥 ∈ ω)) → (∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
4 | 3 | anassrs 397 | . 2 ⊢ ((((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ 𝑃 ∈ Q) ∧ 𝑥 ∈ ω) → (∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
5 | 4 | rexlimdva 2549 | 1 ⊢ (((〈𝐿, 𝑈〉 ∈ P ∧ 𝐴 ∈ 𝐿) ∧ 𝑃 ∈ Q) → (∃𝑥 ∈ ω ∃𝑦 ∈ ω ((𝐴 +Q0 ([〈𝑦, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈((𝑦 +o 2o) +o 𝑥), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈) → ∃𝑗 ∈ ω ((𝐴 +Q0 ([〈𝑗, 1o〉] ~Q0 ·Q0 𝑃)) ∈ 𝐿 ∧ (𝐴 +Q ([〈(𝑗 +o 2o), 1o〉] ~Q ·Q 𝑃)) ∈ 𝑈))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ∃wrex 2417 〈cop 3530 ωcom 4504 (class class class)co 5774 1oc1o 6306 2oc2o 6307 +o coa 6310 [cec 6427 ~Q ceq 7087 Qcnq 7088 +Q cplq 7090 ·Q cmq 7091 ~Q0 ceq0 7094 +Q0 cplq0 7097 ·Q0 cmq0 7098 Pcnp 7099 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-plq0 7235 df-mq0 7236 df-inp 7274 |
This theorem is referenced by: prarloclem 7309 |
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