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Theorem recexprlemm 7565

Description: 𝐵 is inhabited. Lemma for recexpr 7579. (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 7416	. . 3 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prmu 7419	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (2^nd ‘𝐴))
3		recclnq 7333	. . . . . . 7 ⊢ (𝑥 ∈ Q → (*_Q‘𝑥) ∈ Q)
4		nsmallnqq 7353	. . . . . . 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 7335	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Q → (_Q‘(_Q‘𝑥)) = 𝑥)
8	7	eleq1d 2235	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Q → ((_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴) ↔ 𝑥 ∈ (2^nd ‘𝐴)))
9	8	anbi2d 460	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → ((𝑞 <_Q (_Q‘𝑥) ∧ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)) ↔ (𝑞 <_Q (_Q‘𝑥) ∧ 𝑥 ∈ (2^nd ‘𝐴))))
10		breq2 3986	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑞 <_Q 𝑦 ↔ 𝑞 <_Q (_Q‘𝑥)))
11		fveq2 5486	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (_Q‘𝑥) → (_Q‘𝑦) = (_Q‘(_Q‘𝑥)))
12	11	eleq1d 2235	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → ((_Q‘𝑦) ∈ (2^nd ‘𝐴) ↔ (_Q‘(_Q‘𝑥)) ∈ (2^nd ‘𝐴)))
13	10, 12	anbi12d 465	. . . . . . . . . . . 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 2814	. . . . . . . . . . 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 7563	. . . . . . . . 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 1429	. . . . . . 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 2567	. . . . 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 2578	. . 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 7418	. . 3 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑥 ∈ Q 𝑥 ∈ (1^st ‘𝐴))
27		1nq 7307	. . . . . . . 8 ⊢ 1_Q ∈ Q
28		addclnq 7316	. . . . . . . 8 ⊢ (((_Q‘𝑥) ∈ Q ∧ 1_Q ∈ Q) → ((_Q‘𝑥) +_Q 1_Q) ∈ Q)
29	3, 27, 28	sylancl 410	. . . . . . 7 ⊢ (𝑥 ∈ Q → ((*_Q‘𝑥) +_Q 1_Q) ∈ Q)
30		ltaddnq 7348	. . . . . . . 8 ⊢ (((_Q‘𝑥) ∈ Q ∧ 1_Q ∈ Q) → (_Q‘𝑥) <_Q ((*_Q‘𝑥) +_Q 1_Q))
31	3, 27, 30	sylancl 410	. . . . . . 7 ⊢ (𝑥 ∈ Q → (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q))
32		breq2 3986	. . . . . . . 8 ⊢ (𝑟 = ((_Q‘𝑥) +_Q 1_Q) → ((_Q‘𝑥) <_Q 𝑟 ↔ (_Q‘𝑥) <_Q ((_Q‘𝑥) +_Q 1_Q)))
33	32	rspcev 2830	. . . . . . 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 2235	. . . . . . . . . . 11 ⊢ (𝑥 ∈ Q → ((_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴) ↔ 𝑥 ∈ (1^st ‘𝐴)))
37	36	anbi2d 460	. . . . . . . . . 10 ⊢ (𝑥 ∈ Q → (((_Q‘𝑥) <_Q 𝑟 ∧ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)) ↔ ((_Q‘𝑥) <_Q 𝑟 ∧ 𝑥 ∈ (1^st ‘𝐴))))
38		breq1 3985	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → (𝑦 <_Q 𝑟 ↔ (_Q‘𝑥) <_Q 𝑟))
39	11	eleq1d 2235	. . . . . . . . . . . . 13 ⊢ (𝑦 = (_Q‘𝑥) → ((_Q‘𝑦) ∈ (1^st ‘𝐴) ↔ (_Q‘(_Q‘𝑥)) ∈ (1^st ‘𝐴)))
40	38, 39	anbi12d 465	. . . . . . . . . . . 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 2814	. . . . . . . . . . 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 7564	. . . . . . . . 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 1429	. . . . . . 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 2567	. . . . 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 2578	. . 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 1343 ∃wex 1480 ∈ wcel 2136 {cab 2151 ∃wrex 2445 ⟨cop 3579 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 1_Qc1q 7222 +_Q cplq 7223 *_Qcrq 7225 <_Q cltq 7226 Pcnp 7232

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-eprel 4267 df-id 4271 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-1o 6384 df-oadd 6388 df-omul 6389 df-er 6501 df-ec 6503 df-qs 6507 df-ni 7245 df-pli 7246 df-mi 7247 df-lti 7248 df-plpq 7285 df-mpq 7286 df-enq 7288 df-nqqs 7289 df-plqqs 7290 df-mqqs 7291 df-1nqqs 7292 df-rq 7293 df-ltnqqs 7294 df-inp 7407

This theorem is referenced by: recexprlempr 7573

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