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Mirrors > Home > ILE Home > Th. List > prarloclem4 | GIF version |
Description: A slight rearrangement of prarloclem3 7429. Lemma for prarloc 7435. (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 7429 | . . . . 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 1194 | . . . 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 556 | . . 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 398 | . 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 2581 | 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 2135 ∃wrex 2443 〈cop 3573 ωcom 4561 (class class class)co 5836 1oc1o 6368 2oc2o 6369 +o coa 6372 [cec 6490 ~Q ceq 7211 Qcnq 7212 +Q cplq 7214 ·Q cmq 7215 ~Q0 ceq0 7218 +Q0 cplq0 7221 ·Q0 cmq0 7222 Pcnp 7223 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-plq0 7359 df-mq0 7360 df-inp 7398 |
This theorem is referenced by: prarloclem 7433 |
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