Theorem mullocprlem 7931

Description: Calculations for mullocpr 7932. (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 7744	. . . . . . . . 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 7745	. . . . . . . . 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 6267	. . . . . . 7 ⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
12	1, 11	breqtrd 4154	. . . . . 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 7737	. . . . . . . 8 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 ·Q 𝐸) ∈ Q)
16	5, 3, 15	syl2anc 415	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17		ltmnqg 7762	. . . . . . 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 1278	. . . . . 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 7929	. . . . 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 1277	. . . 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 745	. 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 7744	. . . . . . 7 ⊢ ((𝑇 ∈ Q ∧ 𝑈 ∈ Q) → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
36	34, 6, 35	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (𝑇 ·Q 𝑈) = (𝑈 ·Q 𝑇))
37		mullocprlem.tdudr	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅))
38		mulclnq 7737	. . . . . . . . . 10 ⊢ ((𝑇 ∈ Q ∧ 𝑈 ∈ Q) → (𝑇 ·Q 𝑈) ∈ Q)
39	34, 6, 38	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q 𝑈) ∈ Q)
40	13	simprd 114	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Q)
41		ltmnqg 7762	. . . . . . . . 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 1278	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
43	34, 5, 6, 8, 10	caov12d 6265	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
44	43	breq1d 4138	. . . . . . . 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 4152	. . . . 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 7930	. . . . 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 1277	. . . 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 746	. 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 7737	. . . . . . 7 ⊢ ((𝐷 ∈ Q ∧ 𝑈 ∈ Q) → (𝐷 ·Q 𝑈) ∈ Q)
60	4, 59	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61		ltmnqg 7762	. . . . . 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 1278	. . . . 5 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63		mulcomnqg 7744	. . . . . . 7 ⊢ (((𝐷 ·Q 𝑈) ∈ Q ∧ 𝐸 ∈ Q) → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
64	60, 3, 63	syl2anc 415	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝐸) = (𝐸 ·Q (𝐷 ·Q 𝑈)))
65		mulcomnqg 7744	. . . . . . 7 ⊢ (((𝐷 ·Q 𝑈) ∈ Q ∧ 𝑇 ∈ Q) → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
66	60, 34, 65	syl2anc 415	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·Q 𝑈) ·Q 𝑇) = (𝑇 ·Q (𝐷 ·Q 𝑈)))
67	64, 66	breq12d 4141	. . . . 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 7836	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
71		prloc 7852	. . . 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 415	. 2 ⊢ (𝜑 → (𝐸 ∈ (1st ‘𝐵) ∨ 𝑇 ∈ (2nd ‘𝐵)))
74	33, 57, 73	mpjaodan 810	1 ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3711   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079  1^st c1st 6366  2^nd c2nd 6367  Qcnq 7641   ·_Q cmq 7644   <_Q cltq 7646  Pcnp 7652   ·_P cmp 7655

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-mi 7667  df-lti 7668  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714  df-inp 7827  df-imp 7830

This theorem is referenced by:  mullocpr  7932