Theorem mullocprlem 7630

Description: Calculations for mullocpr 7631. (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 7443	. . . . . . . . 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 7444	. . . . . . . . 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 6102	. . . . . . 7 ⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
12	1, 11	breqtrd 4055	. . . . . 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 7436	. . . . . . . 8 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 ·Q 𝐸) ∈ Q)
16	5, 3, 15	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17		ltmnqg 7461	. . . . . . 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 1249	. . . . . 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 7628	. . . . 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 1248	. . . 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 734	. 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 7443	. . . . . . 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 7436	. . . . . . . . . 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 7461	. . . . . . . . 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 1249	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
43	34, 5, 6, 8, 10	caov12d 6100	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
44	43	breq1d 4039	. . . . . . . 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 4053	. . . . 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 7629	. . . . 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 1248	. . . 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 735	. 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 7436	. . . . . . 7 ⊢ ((𝐷 ∈ Q ∧ 𝑈 ∈ Q) → (𝐷 ·Q 𝑈) ∈ Q)
60	4, 59	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61		ltmnqg 7461	. . . . . 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 1249	. . . . 5 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63		mulcomnqg 7443	. . . . . . 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 7443	. . . . . . 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 4042	. . . . 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 7535	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
71		prloc 7551	. . . 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 799	1 ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  ‘cfv 5254  (class class class)co 5918  1^st c1st 6191  2^nd c2nd 6192  Qcnq 7340   ·_Q cmq 7343   <_Q cltq 7345  Pcnp 7351   ·_P cmp 7354

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-mi 7366  df-lti 7367  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-inp 7526  df-imp 7529

This theorem is referenced by:  mullocpr  7631