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Theorem recexprlemm 7614

Description: 𝐵 is inhabited. Lemma for recexpr 7628. (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 7465	. . 3 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prmu 7468	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2^nd ‘𝐴))
3		recclnq 7382	. . . . . . 7 ⊢ (𝑥 ∈ Q → (*_Q‘𝑥) ∈ Q)
4		nsmallnqq 7402	. . . . . . 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 7384	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → (_Q‘(_Q‘𝑥)) = 𝑥)
8	7	eleq1d 2246	. . . . . . . . . . 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 4004	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑞 <_Q 𝑦 ↔ 𝑞 <_Q (_Q‘𝑥)))
11		fveq2 5511	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (_Q‘𝑥) → (_Q‘𝑦) = (_Q‘(_Q‘𝑥)))
12	11	eleq1d 2246	. . . . . . . . . . . . 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 2825	. . . . . . . . . . 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 7612	. . . . . . . . 9 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
19	16, 18	syl6ibr 162	. . . . . . . 8 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (*_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ (1^st ‘𝐵)))
20	19	expcomd 1441	. . . . . . 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 2578	. . . . 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 2589	. . 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 7467	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴))
27		1nq 7356	. . . . . . . 8 ⊢ 1_Q ∈ Q
28		addclnq 7365	. . . . . . . 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 7397	. . . . . . . 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 4004	. . . . . . . 8 ⊢ (𝑟 = ((_Q‘𝑥) +_Q 1_Q) → ((_Q‘𝑥) <_Q 𝑟 ↔ (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q)))
33	32	rspcev 2841	. . . . . . 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 2246	. . . . . . . . . . 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 4003	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑦 <_Q 𝑟 ↔ (_Q‘𝑥) <_Q 𝑟))
39	11	eleq1d 2246	. . . . . . . . . . . . 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 2825	. . . . . . . . . . 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 7613	. . . . . . . . 9 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
45	43, 44	syl6ibr 162	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (((*_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑟 ∈ (2^nd ‘𝐵)))
46	45	expcomd 1441	. . . . . . 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 2578	. . . . 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 2589	. . 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 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 ∃wrex 2456 ⟨cop 3594 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 1^st c1st 6133 2^nd c2nd 6134 Qcnq 7270 1_Qc1q 7271 +_Q cplq 7272 *_Qcrq 7274 <_Q cltq 7275 Pcnp 7281

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-eprel 4286 df-id 4290 df-iord 4363 df-on 4365 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-1o 6411 df-oadd 6415 df-omul 6416 df-er 6529 df-ec 6531 df-qs 6535 df-ni 7294 df-pli 7295 df-mi 7296 df-lti 7297 df-plpq 7334 df-mpq 7335 df-enq 7337 df-nqqs 7338 df-plqqs 7339 df-mqqs 7340 df-1nqqs 7341 df-rq 7342 df-ltnqqs 7343 df-inp 7456

This theorem is referenced by: recexprlempr 7622

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