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Theorem recexprlemex 6609

Description: B is the reciprocal of A. Lemma for recexpr 6610. (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 6607	. . 3 ⊢ (A ∈ P → (1^st ‘(A ·_P B)) ⊆ (1^st ‘1_P))
3	1	recexprlem1ssl 6605	. . 3 ⊢ (A ∈ P → (1^st ‘1_P) ⊆ (1^st ‘(A ·_P B)))
4	2, 3	eqssd 2956	. 2 ⊢ (A ∈ P → (1^st ‘(A ·_P B)) = (1^st ‘1_P))
5	1	recexprlemss1u 6608	. . 3 ⊢ (A ∈ P → (2^nd ‘(A ·_P B)) ⊆ (2^nd ‘1_P))
6	1	recexprlem1ssu 6606	. . 3 ⊢ (A ∈ P → (2^nd ‘1_P) ⊆ (2^nd ‘(A ·_P B)))
7	5, 6	eqssd 2956	. 2 ⊢ (A ∈ P → (2^nd ‘(A ·_P B)) = (2^nd ‘1_P))
8	1	recexprlempr 6604	. . . 4 ⊢ (A ∈ P → B ∈ P)
9		mulclpr 6553	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → (A ·_P B) ∈ P)
10	8, 9	mpdan 398	. . 3 ⊢ (A ∈ P → (A ·_P B) ∈ P)
11		1pr 6535	. . 3 ⊢ 1_P ∈ P
12		preqlu 6455	. . 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 392	. 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 850	1 ⊢ (A ∈ P → (A ·_P B) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 {cab 2023 ⟨cop 3370 class class class wbr 3755 ‘cfv 4845 (class class class)co 5455 1^st c1st 5707 2^nd c2nd 5708 *_Qcrq 6268 <_Q cltq 6269 Pcnp 6275 1_Pc1p 6276 ·_P cmp 6278

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-imp 6452

This theorem is referenced by: recexpr 6610

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