Theorem recexprlemss1u 6734

Description: The upper cut of 𝐴 ·_P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.)

Hypothesis

Ref	Expression
recexpr.1	⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩

Assertion

Ref	Expression
recexprlemss1u	⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

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

Proof of Theorem recexprlemss1u

Dummy variables 𝑞 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		recexpr.1	. . . . . 6 ⊢ 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q‘𝑦) ∈ (2nd ‘𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴))}⟩
2	1	recexprlempr 6730	. . . . 5 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
3		df-imp 6567	. . . . . 6 ⊢ ·P = (𝑦 ∈ P, 𝑤 ∈ P ↦ ⟨{𝑢 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑔 ∈ Q (𝑓 ∈ (1st ‘𝑦) ∧ 𝑔 ∈ (1st ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}, {𝑢 ∈ Q ∣ ∃𝑓 ∈ Q ∃𝑔 ∈ Q (𝑓 ∈ (2nd ‘𝑦) ∧ 𝑔 ∈ (2nd ‘𝑤) ∧ 𝑢 = (𝑓 ·Q 𝑔))}⟩)
4		mulclnq 6474	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
5	3, 4	genpelvu 6611	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
6	2, 5	mpdan 398	. . . 4 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
7	1	recexprlemelu 6721	. . . . . . . 8 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
8		ltrelnq 6463	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
9	8	brel 4392	. . . . . . . . . . . . 13 ⊢ (𝑦 <Q 𝑞 → (𝑦 ∈ Q ∧ 𝑞 ∈ Q))
10	9	simpld 105	. . . . . . . . . . . 12 ⊢ (𝑦 <Q 𝑞 → 𝑦 ∈ Q)
11		prop 6573	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		elprnqu 6580	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
13	11, 12	sylan 267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
14		ltmnqi 6501	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 <Q 𝑞 ∧ 𝑧 ∈ Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
15	14	expcom 109	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ Q → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
16	13, 15	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
17	16	adantr 261	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18		prltlu 6585	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
19	11, 18	syl3an1 1168	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
20	19	3com23 1110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
21	20	3expia 1106	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
22	21	adantr 261	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
23		ltmnqi 6501	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((*Q‘𝑦) <Q 𝑧 ∧ 𝑦 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧))
24	23	expcom 109	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
25	24	adantr 261	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
26		recidnq 6491	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
27	26	adantr 261	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
28		mulcomnqg 6481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
29	27, 28	breq12d 3777	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
30	25, 29	sylibd 138	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
31	30	ancoms 255	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
32	13, 31	sylan 267	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
33	22, 32	syld 40	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑦)))
34	17, 33	anim12d 318	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → ((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦))))
35		ltsonq 6496	. . . . . . . . . . . . . . . 16 ⊢ <Q Or Q
36	35, 8	sotri 4720	. . . . . . . . . . . . . . 15 ⊢ ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
37	36	ancoms 255	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦)) → 1Q <Q (𝑧 ·Q 𝑞))
38	34, 37	syl6 29	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
39	38	exp4b 349	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 ∈ Q → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
40	10, 39	syl5 28	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
41	40	pm2.43d 44	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞))))
42	41	impd 242	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
43	42	exlimdv 1700	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
44	7, 43	syl5bi 141	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45		breq2 3768	. . . . . . . 8 ⊢ (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
46	45	biimprcd 149	. . . . . . 7 ⊢ (1Q <Q (𝑧 ·Q 𝑞) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
47	44, 46	syl6 29	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
48	47	expimpd 345	. . . . 5 ⊢ (𝐴 ∈ P → ((𝑧 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
49	48	rexlimdvv 2439	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
50	6, 49	sylbid 139	. . 3 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51		1pru 6654	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
52	51	abeq2i 2148	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
53	50, 52	syl6ibr 151	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
54	53	ssrdv 2951	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378  1_Qc1q 6379   ·_Q cmq 6381  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389  1_Pc1p 6390   ·_P cmp 6392

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 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

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 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-i1p 6565  df-imp 6567

This theorem is referenced by:  recexprlemex  6735