Theorem recexprlemss1u 7703

Description: The upper cut of 𝐴 ·_P 𝐵 is a subset of the upper cut of one. Lemma for recexpr 7705. (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 7699	. . . . 5 ⊢ (𝐴 ∈ P → 𝐵 ∈ P)
3		df-imp 7536	. . . . . 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 7443	. . . . . 6 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 ·Q 𝑔) ∈ Q)
5	3, 4	genpelvu 7580	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
6	2, 5	mpdan 421	. . . 4 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ ∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞)))
7	1	recexprlemelu 7690	. . . . . . . 8 ⊢ (𝑞 ∈ (2nd ‘𝐵) ↔ ∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)))
8		ltrelnq 7432	. . . . . . . . . . . . . 14 ⊢ <Q ⊆ (Q × Q)
9	8	brel 4715	. . . . . . . . . . . . 13 ⊢ (𝑦 <Q 𝑞 → (𝑦 ∈ Q ∧ 𝑞 ∈ Q))
10	9	simpld 112	. . . . . . . . . . . 12 ⊢ (𝑦 <Q 𝑞 → 𝑦 ∈ Q)
11		prop 7542	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
12		elprnqu 7549	. . . . . . . . . . . . . . . . . 18 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
13	11, 12	sylan 283	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → 𝑧 ∈ Q)
14		ltmnqi 7470	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 <Q 𝑞 ∧ 𝑧 ∈ Q) → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞))
15	14	expcom 116	. . . . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → (𝑦 <Q 𝑞 → (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)))
18		prltlu 7554	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
19	11, 18	syl3an1 1282	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐴 ∈ P ∧ (*Q‘𝑦) ∈ (1st ‘𝐴) ∧ 𝑧 ∈ (2nd ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
20	19	3com23 1211	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴) ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → (*Q‘𝑦) <Q 𝑧)
21	20	3expia 1207	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
22	21	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → (*Q‘𝑦) <Q 𝑧))
23		ltmnqi 7470	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((*Q‘𝑦) <Q 𝑧 ∧ 𝑦 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧))
24	23	expcom 116	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Q → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
25	24	adantr 276	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → (𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧)))
26		recidnq 7460	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
27	26	adantr 276	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q (*Q‘𝑦)) = 1Q)
28		mulcomnqg 7450	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
29	27, 28	breq12d 4046	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑦 ·Q (*Q‘𝑦)) <Q (𝑦 ·Q 𝑧) ↔ 1Q <Q (𝑧 ·Q 𝑦)))
30	25, 29	sylibd 149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
31	30	ancoms 268	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ Q ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
32	13, 31	sylan 283	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) <Q 𝑧 → 1Q <Q (𝑧 ·Q 𝑦)))
33	22, 32	syld 45	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑦)))
34	17, 33	anim12d 335	. . . . . . . . . . . . . 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 7465	. . . . . . . . . . . . . . . 16 ⊢ <Q Or Q
36	35, 8	sotri 5065	. . . . . . . . . . . . . . 15 ⊢ ((1Q <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞)) → 1Q <Q (𝑧 ·Q 𝑞))
37	36	ancoms 268	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ·Q 𝑦) <Q (𝑧 ·Q 𝑞) ∧ 1Q <Q (𝑧 ·Q 𝑦)) → 1Q <Q (𝑧 ·Q 𝑞))
38	34, 37	syl6 33	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) ∧ 𝑦 ∈ Q) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
39	38	exp4b 367	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 ∈ Q → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞)))))
40	10, 39	syl5 32	. . . . . . . . . . 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 50	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑦 <Q 𝑞 → ((*Q‘𝑦) ∈ (1st ‘𝐴) → 1Q <Q (𝑧 ·Q 𝑞))))
42	41	impd 254	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → ((𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
43	42	exlimdv 1833	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (∃𝑦(𝑦 <Q 𝑞 ∧ (*Q‘𝑦) ∈ (1st ‘𝐴)) → 1Q <Q (𝑧 ·Q 𝑞)))
44	7, 43	biimtrid 152	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → 1Q <Q (𝑧 ·Q 𝑞)))
45		breq2 4037	. . . . . . . 8 ⊢ (𝑤 = (𝑧 ·Q 𝑞) → (1Q <Q 𝑤 ↔ 1Q <Q (𝑧 ·Q 𝑞)))
46	45	biimprcd 160	. . . . . . 7 ⊢ (1Q <Q (𝑧 ·Q 𝑞) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
47	44, 46	syl6 33	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑧 ∈ (2nd ‘𝐴)) → (𝑞 ∈ (2nd ‘𝐵) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
48	47	expimpd 363	. . . . 5 ⊢ (𝐴 ∈ P → ((𝑧 ∈ (2nd ‘𝐴) ∧ 𝑞 ∈ (2nd ‘𝐵)) → (𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤)))
49	48	rexlimdvv 2621	. . . 4 ⊢ (𝐴 ∈ P → (∃𝑧 ∈ (2nd ‘𝐴)∃𝑞 ∈ (2nd ‘𝐵)𝑤 = (𝑧 ·Q 𝑞) → 1Q <Q 𝑤))
50	6, 49	sylbid 150	. . 3 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 1Q <Q 𝑤))
51		1pru 7623	. . . 4 ⊢ (2nd ‘1P) = {𝑤 ∣ 1Q <Q 𝑤}
52	51	abeq2i 2307	. . 3 ⊢ (𝑤 ∈ (2nd ‘1P) ↔ 1Q <Q 𝑤)
53	50, 52	imbitrrdi 162	. 2 ⊢ (𝐴 ∈ P → (𝑤 ∈ (2nd ‘(𝐴 ·P 𝐵)) → 𝑤 ∈ (2nd ‘1P)))
54	53	ssrdv 3189	1 ⊢ (𝐴 ∈ P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∃wrex 2476   ⊆ wss 3157  ⟨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 cmq 7350  *_Qcrq 7351   <_Q cltq 7352  Pcnp 7358  1_Pc1p 7359   ·_P cmp 7361

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-po 4331  df-iso 4332  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  df-i1p 7534  df-imp 7536

This theorem is referenced by:  recexprlemex  7704