Theorem mullocprlem 7571

Description: Calculations for mullocpr 7572. (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 112	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Q)
4		mullocprlem.duq	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 ∈ Q ∧ 𝑈 ∈ Q))
5	4	simpld 112	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ Q)
6	4	simprd 114	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Q)
7		mulcomnqg 7384	. . . . . . . . 9 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
8	7	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ Q ∧ 𝑦 ∈ Q)) → (𝑥 ·Q 𝑦) = (𝑦 ·Q 𝑥))
9		mulassnqg 7385	. . . . . . . . 9 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → ((𝑥 ·Q 𝑦) ·Q 𝑧) = (𝑥 ·Q (𝑦 ·Q 𝑧)))
10	9	adantl 277	. . . . . . . 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 6060	. . . . . . 7 ⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
12	1, 11	breqtrd 4031	. . . . . 6 ⊢ (𝜑 → (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸)))
13		mullocprlem.qr	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∈ Q ∧ 𝑅 ∈ Q))
14	13	simpld 112	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Q)
15		mulclnq 7377	. . . . . . . 8 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 ·Q 𝐸) ∈ Q)
16	5, 3, 15	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17		ltmnqg 7402	. . . . . . 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 1238	. . . . . 6 ⊢ (𝜑 → (𝑄 <Q (𝐷 ·Q 𝐸) ↔ (𝑈 ·Q 𝑄) <Q (𝑈 ·Q (𝐷 ·Q 𝐸))))
19	12, 18	mpbird 167	. . . . 5 ⊢ (𝜑 → 𝑄 <Q (𝐷 ·Q 𝐸))
20	19	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝑄 <Q (𝐷 ·Q 𝐸))
21		mullocprlem.ab	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐵 ∈ P))
22	21	simpld 112	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ P)
23		mullocprlem.du	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ∈ (1st ‘𝐴) ∧ 𝑈 ∈ (2nd ‘𝐴)))
24	23	simpld 112	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (1st ‘𝐴))
25	22, 24	jca 306	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)))
26	25	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝐴 ∈ P ∧ 𝐷 ∈ (1st ‘𝐴)))
27	21	simprd 114	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ P)
28	27	anim1i 340	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝐵 ∈ P ∧ 𝐸 ∈ (1st ‘𝐵)))
29	14	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → 𝑄 ∈ Q)
30		mulnqprl 7569	. . . . 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 1237	. . . 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 733	. 2 ⊢ ((𝜑 ∧ 𝐸 ∈ (1st ‘𝐵)) → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))
34	2	simprd 114	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ Q)
35		mulcomnqg 7384	. . . . . . 7 ⊢ ((𝑇 ∈ Q ∧ 𝑈 ∈ Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
36	34, 6, 35	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37		mullocprlem.tdudr	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38		mulclnq 7377	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Q ∧ 𝑈 ∈ Q) → (𝑇 ·Q 𝑈) ∈ Q)
39	34, 6, 38	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
40	13	simprd 114	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Q)
41		ltmnqg 7402	. . . . . . . . 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 1238	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
43	34, 5, 6, 8, 10	caov12d 6058	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
44	43	breq1d 4015	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅) ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
45	42, 44	bitr4d 191	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
46	37, 45	mpbird 167	. . . . . 6 ⊢ (𝜑 → (𝑇 ·Q 𝑈) <Q 𝑅)
47	36, 46	eqbrtrrd 4029	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝑇) <Q 𝑅)
48	47	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝑈 ·Q 𝑇) <Q 𝑅)
49	23	simprd 114	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (2nd ‘𝐴))
50	22, 49	jca 306	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
51	50	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝐴 ∈ P ∧ 𝑈 ∈ (2nd ‘𝐴)))
52	27	anim1i 340	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → (𝐵 ∈ P ∧ 𝑇 ∈ (2nd ‘𝐵)))
53	40	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 ∈ (2nd ‘𝐵)) → 𝑅 ∈ Q)
54		mulnqpru 7570	. . . . 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 1237	. . . 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 734	. 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 7377	. . . . . . 7 ⊢ ((𝐷 ∈ Q ∧ 𝑈 ∈ Q) → (𝐷 ·Q 𝑈) ∈ Q)
60	4, 59	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61		ltmnqg 7402	. . . . . 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 1238	. . . . 5 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63		mulcomnqg 7384	. . . . . . 7 ⊢ (((𝐷 ·Q 𝑈) ∈ Q ∧ 𝐸 ∈ Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
64	60, 3, 63	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65		mulcomnqg 7384	. . . . . . 7 ⊢ (((𝐷 ·Q 𝑈) ∈ Q ∧ 𝑇 ∈ Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
66	60, 34, 65	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
67	64, 66	breq12d 4018	. . . . 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 188	. . . 4 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ (𝐸 ·Q (𝐷 ·Q 𝑈)) <Q (𝑇 ·Q (𝐷 ·Q 𝑈))))
69	58, 68	mpbird 167	. . 3 ⊢ (𝜑 → 𝐸 <Q 𝑇)
70		prop 7476	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
71		prloc 7492	. . . 4 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐸 <Q 𝑇) → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
72	70, 71	sylan 283	. . 3 ⊢ ((𝐵 ∈ P ∧ 𝐸 <Q 𝑇) → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
73	27, 69, 72	syl2anc 411	. 2 ⊢ (𝜑 → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
74	33, 57, 73	mpjaodan 798	1 ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ⟨cop 3597   class class class wbr 4005  ‘cfv 5218  (class class class)co 5877  1^st c1st 6141  2^nd c2nd 6142  Qcnq 7281   ·_Q cmq 7284   <_Q cltq 7286  Pcnp 7292   ·_P cmp 7295

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-mi 7307  df-lti 7308  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-imp 7470

This theorem is referenced by:  mullocpr  7572