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Theorem recexprlemex 6793

Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6794. (Contributed by Jim Kingdon, 27-Dec-2019.)

**Hypothesis**
Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩

**Assertion**
Ref	Expression
recexprlemex	⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦

**Proof of Theorem recexprlemex**
Step	Hyp	Ref	Expression
1		recexpr.1	. . . 4 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
2	1	recexprlemss1l 6791	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) ⊆ (1^st ‘1_P))
3	1	recexprlem1ssl 6789	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘1_P) ⊆ (1^st ‘(𝐴 ·_P 𝐵)))
4	2, 3	eqssd 2990	. 2 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P))
5	1	recexprlemss1u 6792	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) ⊆ (2^nd ‘1_P))
6	1	recexprlem1ssu 6790	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘1_P) ⊆ (2^nd ‘(𝐴 ·_P 𝐵)))
7	5, 6	eqssd 2990	. 2 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))
8	1	recexprlempr 6788	. . . 4 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
9		mulclpr 6728	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
10	8, 9	mpdan 406	. . 3 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) ∈ P)
11		1pr 6710	. . 3 ⊢ 1_P ∈ P
12		preqlu 6628	. . 3 ⊢ (((𝐴 ·_P 𝐵) ∈ P ∧ 1_P ∈ P) → ((𝐴 ·_P 𝐵) = 1_P ↔ ((1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P) ∧ (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))))
13	10, 11, 12	sylancl 398	. 2 ⊢ (𝐴 ∈ P → ((𝐴 ·_P 𝐵) = 1_P ↔ ((1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P) ∧ (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))))
14	4, 7, 13	mpbir2and 862	1 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 101 ↔ wb 102 = wceq 1259 ∃wex 1397 ∈ wcel 1409 {cab 2042 ⟨cop 3406 class class class wbr 3792 ‘cfv 4930 (class class class)co 5540 1^st c1st 5793 2^nd c2nd 5794 *_Qcrq 6440 <_Q cltq 6441 Pcnp 6447 1_Pc1p 6448 ·_P cmp 6450

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 103 ax-ia2 104 ax-ia3 105 ax-in1 554 ax-in2 555 ax-io 640 ax-5 1352 ax-7 1353 ax-gen 1354 ax-ie1 1398 ax-ie2 1399 ax-8 1411 ax-10 1412 ax-11 1413 ax-i12 1414 ax-bndl 1415 ax-4 1416 ax-13 1420 ax-14 1421 ax-17 1435 ax-i9 1439 ax-ial 1443 ax-i5r 1444 ax-ext 2038 ax-coll 3900 ax-sep 3903 ax-nul 3911 ax-pow 3955 ax-pr 3972 ax-un 4198 ax-setind 4290 ax-iinf 4339

This theorem depends on definitions: df-bi 114 df-dc 754 df-3or 897 df-3an 898 df-tru 1262 df-fal 1265 df-nf 1366 df-sb 1662 df-eu 1919 df-mo 1920 df-clab 2043 df-cleq 2049 df-clel 2052 df-nfc 2183 df-ne 2221 df-ral 2328 df-rex 2329 df-reu 2330 df-rab 2332 df-v 2576 df-sbc 2788 df-csb 2881 df-dif 2948 df-un 2950 df-in 2952 df-ss 2959 df-nul 3253 df-pw 3389 df-sn 3409 df-pr 3410 df-op 3412 df-uni 3609 df-int 3644 df-iun 3687 df-br 3793 df-opab 3847 df-mpt 3848 df-tr 3883 df-eprel 4054 df-id 4058 df-po 4061 df-iso 4062 df-iord 4131 df-on 4133 df-suc 4136 df-iom 4342 df-xp 4379 df-rel 4380 df-cnv 4381 df-co 4382 df-dm 4383 df-rn 4384 df-res 4385 df-ima 4386 df-iota 4895 df-fun 4932 df-fn 4933 df-f 4934 df-f1 4935 df-fo 4936 df-f1o 4937 df-fv 4938 df-ov 5543 df-oprab 5544 df-mpt2 5545 df-1st 5795 df-2nd 5796 df-recs 5951 df-irdg 5988 df-1o 6032 df-2o 6033 df-oadd 6036 df-omul 6037 df-er 6137 df-ec 6139 df-qs 6143 df-ni 6460 df-pli 6461 df-mi 6462 df-lti 6463 df-plpq 6500 df-mpq 6501 df-enq 6503 df-nqqs 6504 df-plqqs 6505 df-mqqs 6506 df-1nqqs 6507 df-rq 6508 df-ltnqqs 6509 df-enq0 6580 df-nq0 6581 df-0nq0 6582 df-plq0 6583 df-mq0 6584 df-inp 6622 df-i1p 6623 df-imp 6625

This theorem is referenced by: recexpr 6794

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