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Theorem recexprlemm 7586

Description: 𝐵 is inhabited. Lemma for recexpr 7600. (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 7437	. . 3 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prmu 7440	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2^nd ‘𝐴))
3		recclnq 7354	. . . . . . 7 ⊢ (𝑥 ∈ Q → (*_Q‘𝑥) ∈ Q)
4		nsmallnqq 7374	. . . . . . 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 274	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (2^nd ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 <_Q (*_Q‘𝑥))
7		recrecnq 7356	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → (_Q‘(_Q‘𝑥)) = 𝑥)
8	7	eleq1d 2239	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Q → ((_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴) ↔ 𝑥 ∈ (2^nd ‘𝐴)))
9	8	anbi2d 461	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (_Q‘𝑥) ∧ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)) ↔ (𝑞 <_Q (_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴))))
10		breq2 3993	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑞 <_Q 𝑦 ↔ 𝑞 <_Q (_Q‘𝑥)))
11		fveq2 5496	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (_Q‘𝑥) → (_Q‘𝑦) = (_Q‘(_Q‘𝑥)))
12	11	eleq1d 2239	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → ((_Q‘𝑦) ∈ (2^nd ‘𝐴) ↔ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)))
13	10, 12	anbi12d 470	. . . . . . . . . . . 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 2818	. . . . . . . . . . 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 169	. . . . . . . . 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 7584	. . . . . . . . 9 ⊢ (𝑞 ∈ (1^st ‘𝐵) ↔ ∃𝑦(𝑞 <_Q 𝑦 ∧ (*_Q‘𝑦) ∈ (2^nd ‘𝐴)))
19	16, 18	syl6ibr 161	. . . . . . . 8 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (*_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ (1^st ‘𝐵)))
20	19	expcomd 1434	. . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ∈ (2^nd ‘𝐴) → (𝑞 <_Q (*_Q‘𝑥) → 𝑞 ∈ (1^st ‘𝐵))))
21	20	imp 123	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (2^nd ‘𝐴)) → (𝑞 <_Q (*_Q‘𝑥) → 𝑞 ∈ (1^st ‘𝐵)))
22	21	reximdv 2571	. . . . 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 2582	. . 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 7439	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴))
27		1nq 7328	. . . . . . . 8 ⊢ 1_Q ∈ Q
28		addclnq 7337	. . . . . . . 8 ⊢ (((_Q‘𝑥) ∈ Q ∧ 1_Q ∈ Q) → ((_Q‘𝑥) +_Q 1_Q) ∈ Q)
29	3, 27, 28	sylancl 411	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*_Q‘𝑥) +_Q 1_Q) ∈ Q)
30		ltaddnq 7369	. . . . . . . 8 ⊢ (((_Q‘𝑥) ∈ Q ∧ 1_Q ∈ Q) → (_Q‘𝑥) <_Q ((*_Q‘𝑥) +_Q 1_Q))
31	3, 27, 30	sylancl 411	. . . . . . 7 ⊢ (𝑥 ∈ Q → (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q))
32		breq2 3993	. . . . . . . 8 ⊢ (𝑟 = ((_Q‘𝑥) +_Q 1_Q) → ((_Q‘𝑥) <_Q 𝑟 ↔ (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q)))
33	32	rspcev 2834	. . . . . . 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 409	. . . . . 6 ⊢ (𝑥 ∈ Q → ∃𝑟 ∈ Q (*_Q‘𝑥) <_Q 𝑟)
35	34	adantr 274	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑟 ∈ Q (*_Q‘𝑥) <_Q 𝑟)
36	7	eleq1d 2239	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Q → ((_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴) ↔ 𝑥 ∈ (1^st ‘𝐴)))
37	36	anbi2d 461	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)) ↔ ((_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴))))
38		breq1 3992	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑦 <_Q 𝑟 ↔ (_Q‘𝑥) <_Q 𝑟))
39	11	eleq1d 2239	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → ((_Q‘𝑦) ∈ (1^st ‘𝐴) ↔ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)))
40	38, 39	anbi12d 470	. . . . . . . . . . . 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 2818	. . . . . . . . . . 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 169	. . . . . . . . 9 ⊢ (𝑥 ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴)) → ∃𝑦(𝑦 <_Q 𝑟 ∧ (_Q‘𝑦) ∈ (1^st ‘𝐴))))
44	17	recexprlemelu 7585	. . . . . . . . 9 ⊢ (𝑟 ∈ (2^nd ‘𝐵) ↔ ∃𝑦(𝑦 <_Q 𝑟 ∧ (*_Q‘𝑦) ∈ (1^st ‘𝐴)))
45	43, 44	syl6ibr 161	. . . . . . . 8 ⊢ (𝑥 ∈ Q → (((*_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴)) → 𝑟 ∈ (2^nd ‘𝐵)))
46	45	expcomd 1434	. . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ∈ (1^st ‘𝐴) → ((*_Q‘𝑥) <_Q 𝑟 → 𝑟 ∈ (2^nd ‘𝐵))))
47	46	imp 123	. . . . . 6 ⊢ ((𝑥 ∈ Q ∧ 𝑥 ∈ (1^st ‘𝐴)) → ((*_Q‘𝑥) <_Q 𝑟 → 𝑟 ∈ (2^nd ‘𝐵)))
48	47	reximdv 2571	. . . . 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 2582	. . 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 304	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 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 ∃wrex 2449 ⟨cop 3586 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 1^st c1st 6117 2^nd c2nd 6118 Qcnq 7242 1_Qc1q 7243 +_Q cplq 7244 *_Qcrq 7246 <_Q cltq 7247 Pcnp 7253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-inp 7428

This theorem is referenced by: recexprlempr 7594

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