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Theorem recexprlemm 7691

Description: 𝐵 is inhabited. Lemma for recexpr 7705. (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
recexprlemm	⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵)))

Distinct variable groups: 𝑟,𝑞,𝑥,𝑦,𝐴 𝐵,𝑞,𝑟,𝑥,𝑦

**Proof of Theorem recexprlemm**
Step	Hyp	Ref	Expression
1		prop 7542	. . 3 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prmu 7545	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2^nd ‘𝐴))
3		recclnq 7459	. . . . . . 7 ⊢ (𝑥 ∈ Q → (*_Q‘𝑥) ∈ Q)
4		nsmallnqq 7479	. . . . . . 7 ⊢ ((_Q‘𝑥) ∈ Q → ∃𝑞 ∈ Q 𝑞 <_Q (_Q‘𝑥))
5	3, 4	syl 14	. . . . . 6 ⊢ (𝑥 ∈ Q → ∃𝑞 ∈ Q 𝑞 <_Q (*_Q‘𝑥))
6	5	adantr 276	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (2^nd ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 <_Q (*_Q‘𝑥))
7		recrecnq 7461	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → (_Q‘(_Q‘𝑥)) = 𝑥)
8	7	eleq1d 2265	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Q → ((_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴) ↔ 𝑥 ∈ (2^nd ‘𝐴)))
9	8	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (_Q‘𝑥) ∧ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)) ↔ (𝑞 <_Q (_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴))))
10		breq2 4037	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑞 <_Q 𝑦 ↔ 𝑞 <_Q (_Q‘𝑥)))
11		fveq2 5558	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (_Q‘𝑥) → (_Q‘𝑦) = (_Q‘(_Q‘𝑥)))
12	11	eleq1d 2265	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → ((_Q‘𝑦) ∈ (2^nd ‘𝐴) ↔ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)))
13	10, 12	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑦 = (_Q‘𝑥) → ((𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴)) ↔ (𝑞 <_Q (_Q‘𝑥) ∧ (_Q‘(*_Q‘𝑥)) ∈ (2^nd ‘𝐴))))
14	13	spcegv 2852	. . . . . . . . . . 11 ⊢ ((_Q‘𝑥) ∈ Q → ((𝑞 <_Q (_Q‘𝑥) ∧ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴))))
15	3, 14	syl 14	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (_Q‘𝑥) ∧ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
16	9, 15	sylbird 170	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → ∃𝑦(𝑞 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))))
17		recexpr.1	. . . . . . . . . 10 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <_Q 𝑦 ∧ (_Q‘𝑦) ∈ (2^nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <_Q 𝑥 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))}⟩
18	17	recexprlemell 7689	. . . . . . . . 9 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
19	16, 18	imbitrrdi 162	. . . . . . . 8 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (*_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ (1^st ‘𝐵)))
20	19	expcomd 1452	. . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ∈ (2^nd ‘𝐴) → (𝑞 <_Q (*_Q‘𝑥) → 𝑞 ∈ (1^st ‘𝐵))))
21	20	imp 124	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (2^nd ‘𝐴)) → (𝑞 <_Q (*_Q‘𝑥) → 𝑞 ∈ (1^st ‘𝐵)))
22	21	reximdv 2598	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (2^nd ‘𝐴)) → (∃𝑞 ∈ Q 𝑞 <_Q (*_Q‘𝑥) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵)))
23	6, 22	mpd 13	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (2^nd ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵))
24	23	rexlimiva 2609	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (2^nd ‘𝐴) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵))
25	1, 2, 24	3syl 17	. 2 ⊢ (𝐴 ∈ P → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵))
26		prml 7544	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴))
27		1nq 7433	. . . . . . . 8 ⊢ 1_Q ∈ Q
28		addclnq 7442	. . . . . . . 8 ⊢ (((_Q‘𝑥) ∈ Q ∧ 1_Q ∈ Q) → ((_Q‘𝑥) +_Q 1_Q) ∈ Q)
29	3, 27, 28	sylancl 413	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*_Q‘𝑥) +_Q 1_Q) ∈ Q)
30		ltaddnq 7474	. . . . . . . 8 ⊢ (((_Q‘𝑥) ∈ Q ∧ 1_Q ∈ Q) → (_Q‘𝑥) <_Q ((*_Q‘𝑥) +_Q 1_Q))
31	3, 27, 30	sylancl 413	. . . . . . 7 ⊢ (𝑥 ∈ Q → (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q))
32		breq2 4037	. . . . . . . 8 ⊢ (𝑟 = ((_Q‘𝑥) +_Q 1_Q) → ((_Q‘𝑥) <_Q 𝑟 ↔ (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q)))
33	32	rspcev 2868	. . . . . . 7 ⊢ ((((_Q‘𝑥) +_Q 1_Q) ∈ Q ∧ (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q)) → ∃𝑟 ∈ Q (_Q‘𝑥) <_Q 𝑟)
34	29, 31, 33	syl2anc 411	. . . . . 6 ⊢ (𝑥 ∈ Q → ∃𝑟 ∈ Q (*_Q‘𝑥) <_Q 𝑟)
35	34	adantr 276	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑟 ∈ Q (*_Q‘𝑥) <_Q 𝑟)
36	7	eleq1d 2265	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Q → ((_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴) ↔ 𝑥 ∈ (1^st ‘𝐴)))
37	36	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)) ↔ ((_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴))))
38		breq1 4036	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑦 <_Q 𝑟 ↔ (_Q‘𝑥) <_Q 𝑟))
39	11	eleq1d 2265	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → ((_Q‘𝑦) ∈ (1^st ‘𝐴) ↔ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)))
40	38, 39	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑦 = (_Q‘𝑥) → ((𝑦 <_Q 𝑟 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴)) ↔ ((_Q‘𝑥) <_Q 𝑟 ∧ (_Q‘(*_Q‘𝑥)) ∈ (1^st ‘𝐴))))
41	40	spcegv 2852	. . . . . . . . . . 11 ⊢ ((_Q‘𝑥) ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴))))
42	3, 41	syl 14	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑟 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
43	37, 42	sylbird 170	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑟 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
44	17	recexprlemelu 7690	. . . . . . . . 9 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
45	43, 44	imbitrrdi 162	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (((*_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑟 ∈ (2^nd ‘𝐵)))
46	45	expcomd 1452	. . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ∈ (1^st ‘𝐴) → ((*_Q‘𝑥) <_Q 𝑟 → 𝑟 ∈ (2^nd ‘𝐵))))
47	46	imp 124	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴)) → ((*_Q‘𝑥) <_Q 𝑟 → 𝑟 ∈ (2^nd ‘𝐵)))
48	47	reximdv 2598	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴)) → (∃𝑟 ∈ Q (*_Q‘𝑥) <_Q 𝑟 → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵)))
49	35, 48	mpd 13	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))
50	49	rexlimiva 2609	. . 3 ⊢ (∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))
51	1, 26, 50	3syl 17	. 2 ⊢ (𝐴 ∈ P → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵))
52	25, 51	jca 306	1 ⊢ (𝐴 ∈ P → (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘𝐵) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∃wex 1506 ∈ wcel 2167 {cab 2182 ∃wrex 2476 ⟨cop 3625 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 1_Qc1q 7348 +_Q cplq 7349 *_Qcrq 7351 <_Q cltq 7352 Pcnp 7358

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-eprel 4324 df-id 4328 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-1o 6474 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-pli 7372 df-mi 7373 df-lti 7374 df-plpq 7411 df-mpq 7412 df-enq 7414 df-nqqs 7415 df-plqqs 7416 df-mqqs 7417 df-1nqqs 7418 df-rq 7419 df-ltnqqs 7420 df-inp 7533

This theorem is referenced by: recexprlempr 7699

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