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Theorem recexprlemex 6678

Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6679. (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 6676	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) ⊆ (1^st ‘1_P))
3	1	recexprlem1ssl 6674	. . 3 ⊢ (𝐴 ∈ P → (1^st ‘1_P) ⊆ (1^st ‘(𝐴 ·_P 𝐵)))
4	2, 3	eqssd 2959	. 2 ⊢ (𝐴 ∈ P → (1^st ‘(𝐴 ·_P 𝐵)) = (1^st ‘1_P))
5	1	recexprlemss1u 6677	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) ⊆ (2^nd ‘1_P))
6	1	recexprlem1ssu 6675	. . 3 ⊢ (𝐴 ∈ P → (2^nd ‘1_P) ⊆ (2^nd ‘(𝐴 ·_P 𝐵)))
7	5, 6	eqssd 2959	. 2 ⊢ (𝐴 ∈ P → (2^nd ‘(𝐴 ·_P 𝐵)) = (2^nd ‘1_P))
8	1	recexprlempr 6673	. . . 4 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
9		mulclpr 6613	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)
10	8, 9	mpdan 398	. . 3 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) ∈ P)
11		1pr 6595	. . 3 ⊢ 1_P ∈ P
12		preqlu 6513	. . 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 851	1 ⊢ (𝐴 ∈ P → (𝐴 ·_P 𝐵) = 1_P)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 ∃wex 1381 ∈ wcel 1393 {cab 2026 ⟨cop 3375 class class class wbr 3760 ‘cfv 4863 (class class class)co 5473 1^st c1st 5726 2^nd c2nd 5727 *_Qcrq 6325 <_Q cltq 6326 Pcnp 6332 1_Pc1p 6333 ·_P cmp 6335

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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3868 ax-sep 3871 ax-nul 3879 ax-pow 3923 ax-pr 3940 ax-un 4141 ax-setind 4230 ax-iinf 4272

This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2308 df-rex 2309 df-reu 2310 df-rab 2312 df-v 2556 df-sbc 2762 df-csb 2850 df-dif 2917 df-un 2919 df-in 2921 df-ss 2928 df-nul 3222 df-pw 3358 df-sn 3378 df-pr 3379 df-op 3381 df-uni 3577 df-int 3612 df-iun 3655 df-br 3761 df-opab 3815 df-mpt 3816 df-tr 3851 df-eprel 4022 df-id 4026 df-po 4029 df-iso 4030 df-iord 4074 df-on 4076 df-suc 4079 df-iom 4275 df-xp 4312 df-rel 4313 df-cnv 4314 df-co 4315 df-dm 4316 df-rn 4317 df-res 4318 df-ima 4319 df-iota 4828 df-fun 4865 df-fn 4866 df-f 4867 df-f1 4868 df-fo 4869 df-f1o 4870 df-fv 4871 df-ov 5476 df-oprab 5477 df-mpt2 5478 df-1st 5728 df-2nd 5729 df-recs 5881 df-irdg 5918 df-1o 5962 df-2o 5963 df-oadd 5966 df-omul 5967 df-er 6065 df-ec 6067 df-qs 6071 df-ni 6345 df-pli 6346 df-mi 6347 df-lti 6348 df-plpq 6385 df-mpq 6386 df-enq 6388 df-nqqs 6389 df-plqqs 6390 df-mqqs 6391 df-1nqqs 6392 df-rq 6393 df-ltnqqs 6394 df-enq0 6465 df-nq0 6466 df-0nq0 6467 df-plq0 6468 df-mq0 6469 df-inp 6507 df-i1p 6508 df-imp 6510

This theorem is referenced by: recexpr 6679

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