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Mirrors > Home > ILE Home > Th. List > recexprlemex | GIF version |
Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6828. (Contributed by Jim Kingdon, 27-Dec-2019.) |
Ref | Expression |
---|---|
recexpr.1 | ⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 |
Ref | Expression |
---|---|
recexprlemex | ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recexpr.1 | . . . 4 ⊢ 𝐵 = 〈{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}〉 | |
2 | 1 | recexprlemss1l 6825 | . . 3 ⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P)) |
3 | 1 | recexprlem1ssl 6823 | . . 3 ⊢ (𝐴 ∈ P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵))) |
4 | 2, 3 | eqssd 3016 | . 2 ⊢ (𝐴 ∈ P → (1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P)) |
5 | 1 | recexprlemss1u 6826 | . . 3 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P)) |
6 | 1 | recexprlem1ssu 6824 | . . 3 ⊢ (𝐴 ∈ P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵))) |
7 | 5, 6 | eqssd 3016 | . 2 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P)) |
8 | 1 | recexprlempr 6822 | . . . 4 ⊢ (𝐴 ∈ P → 𝐵 ∈ P) |
9 | mulclpr 6762 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·P 𝐵) ∈ P) | |
10 | 8, 9 | mpdan 412 | . . 3 ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) ∈ P) |
11 | 1pr 6744 | . . 3 ⊢ 1P ∈ P | |
12 | preqlu 6662 | . . 3 ⊢ (((𝐴 ·P 𝐵) ∈ P ∧ 1P ∈ P) → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P)))) | |
13 | 10, 11, 12 | sylancl 404 | . 2 ⊢ (𝐴 ∈ P → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P)))) |
14 | 4, 7, 13 | mpbir2and 885 | 1 ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 〈cop 3401 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 1st c1st 5785 2nd c2nd 5786 *Qcrq 6474 <Q cltq 6475 Pcnp 6481 1Pc1p 6482 ·P cmp 6484 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-imp 6659 |
This theorem is referenced by: recexpr 6828 |
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