Theorem mullocprlem 7378

Description: Calculations for mullocpr 7379. (Contributed by Jim Kingdon, 10-Dec-2019.)

Hypotheses

Ref	Expression
mullocprlem.ab	⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
mullocprlem.uqedu	⊢ (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))
mullocprlem.edutdu	⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))
mullocprlem.tdudr	⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
mullocprlem.qr	⊢ (𝜑 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
mullocprlem.duq	⊢ (𝜑 → (𝐷 ∈ Q ∧ 𝑈 ∈ Q))
mullocprlem.du	⊢ (𝜑 → (𝐷 ∈ (1st ‘𝐴) ∧ 𝑈 ∈ (2nd ‘𝐴)))
mullocprlem.et	⊢ (𝜑 → (𝐸 ∈ Q ∧ 𝑇 ∈ Q))

Assertion

Ref	Expression
mullocprlem	⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Proof of Theorem mullocprlem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mullocprlem.uqedu	. . . . . . 7 ⊢ (𝜑 → (𝑈 ·Q 𝑄) <Q (𝐸 ·Q (𝐷 ·Q 𝑈)))
2		mullocprlem.et	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 ∈ Q ∧ 𝑇 ∈ Q))
3	2	simpld 111	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Q)
4		mullocprlem.duq	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ Q ∧ 𝑈 ∈ Q))
5	4	simpld 111	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Q)
6	4	simprd 113	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Q)
7		mulcomnqg 7191	. . . . . . . . 9 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
8	7	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9		mulassnqg 7192	. . . . . . . . 9 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
10	9	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
11	3, 5, 6, 8, 10	caov13d 5954	. . . . . . 7 ⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
12	1, 11	breqtrd 3954	. . . . . 6 ⊢ (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13		mullocprlem.qr	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
14	13	simpld 111	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Q)
15		mulclnq 7184	. . . . . . . 8 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 ·Q 𝐸) ∈ Q)
16	5, 3, 15	syl2anc 408	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17		ltmnqg 7209	. . . . . . 7 ⊢ ((𝑄 ∈ Q ∧ (𝐷 ·Q 𝐸) ∈ Q ∧ 𝑈 ∈ Q) → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
18	14, 16, 6, 17	syl3anc 1216	. . . . . 6 ⊢ (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
19	12, 18	mpbird 166	. . . . 5 ⊢ (𝜑 → 𝑄 <Q (𝐷 ·Q 𝐸))
20	19	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21		mullocprlem.ab	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
22	21	simpld 111	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ P)
23		mullocprlem.du	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ (1st ‘𝐴) ∧ 𝑈 ∈ (2nd ‘𝐴)))
24	23	simpld 111	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
25	22, 24	jca 304	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)))
26	25	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)))
27	21	simprd 113	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
28	27	anim1i 338	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝐵 ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)))
29	14	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝑄 ∈ Q)
30		mulnqprl 7376	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐸 ∈ (1st ‘𝐵))) ∧ 𝑄 ∈ Q) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
31	26, 28, 29, 30	syl21anc 1215	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝑄 <Q (𝐷 ·Q 𝐸) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵))))
32	20, 31	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)))
33	32	orcd 722	. 2 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
34	2	simprd 113	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ Q)
35		mulcomnqg 7191	. . . . . . 7 ⊢ ((𝑇 ∈ Q ∧ 𝑈 ∈ Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
36	34, 6, 35	syl2anc 408	. . . . . 6 ⊢ (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37		mullocprlem.tdudr	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38		mulclnq 7184	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Q ∧ 𝑈 ∈ Q) → (𝑇 ·Q 𝑈) ∈ Q)
39	34, 6, 38	syl2anc 408	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
40	13	simprd 113	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Q)
41		ltmnqg 7209	. . . . . . . . 9 ⊢ (((𝑇 ·Q 𝑈) ∈ Q ∧ 𝑅 ∈ Q ∧ 𝐷 ∈ Q) → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
42	39, 40, 5, 41	syl3anc 1216	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
43	34, 5, 6, 8, 10	caov12d 5952	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
44	43	breq1d 3939	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
45	42, 44	bitr4d 190	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
46	37, 45	mpbird 166	. . . . . 6 ⊢ (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
47	36, 46	eqbrtrrd 3952	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
48	47	adantr 274	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
49	23	simprd 113	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
50	22, 49	jca 304	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
51	50	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
52	27	anim1i 338	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)))
53	40	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑅 ∈ Q)
54		mulnqpru 7377	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵))) ∧ 𝑅 ∈ Q) → ((𝑈 ·Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
55	51, 52, 53, 54	syl21anc 1215	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → ((𝑈 ·Q 𝑇) <Q 𝑅 → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
56	48, 55	mpd 13	. . 3 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵)))
57	56	olcd 723	. 2 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
58		mullocprlem.edutdu	. . . 4 ⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈)))
59		mulclnq 7184	. . . . . . 7 ⊢ ((𝐷 ∈ Q ∧ 𝑈 ∈ Q) → (𝐷 ·Q 𝑈) ∈ Q)
60	4, 59	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61		ltmnqg 7209	. . . . . 6 ⊢ ((𝐸 ∈ Q ∧ 𝑇 ∈ Q ∧ (𝐷 ·Q 𝑈) ∈ Q) → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
62	3, 34, 60, 61	syl3anc 1216	. . . . 5 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63		mulcomnqg 7191	. . . . . . 7 ⊢ (((𝐷 ·Q 𝑈) ∈ Q ∧ 𝐸 ∈ Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
64	60, 3, 63	syl2anc 408	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65		mulcomnqg 7191	. . . . . . 7 ⊢ (((𝐷 ·Q 𝑈) ∈ Q ∧ 𝑇 ∈ Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
66	60, 34, 65	syl2anc 408	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
67	64, 66	breq12d 3942	. . . . 5 ⊢ (𝜑 → (((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇) ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
68	62, 67	bitrd 187	. . . 4 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
69	58, 68	mpbird 166	. . 3 ⊢ (𝜑 → 𝐸 <Q 𝑇)
70		prop 7283	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
71		prloc 7299	. . . 4 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 <Q 𝑇) → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
72	70, 71	sylan 281	. . 3 ⊢ ((𝐵 ∈ P ∧ 𝐸 <Q 𝑇) → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
73	27, 69, 72	syl2anc 408	. 2 ⊢ (𝜑 → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
74	33, 57, 73	mpjaodan 787	1 ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1^st c1st 6036  2^nd c2nd 6037  Qcnq 7088   ·_Q cmq 7091   <_Q cltq 7093  Pcnp 7099   ·_P cmp 7102

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-mi 7114  df-lti 7115  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-inp 7274  df-imp 7277

This theorem is referenced by:  mullocpr  7379