Theorem mullocprlem 7690

Description: Calculations for mullocpr 7691. (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 7503	. . . . . . . . 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 7504	. . . . . . . . 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 6137	. . . . . . 7 ⊢ (𝜑 → (𝐸 ·Q (𝐷 ·Q 𝑈)) = (𝑈 ·Q (𝐷 ·Q 𝐸)))
12	1, 11	breqtrd 4073	. . . . . 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 7496	. . . . . . . 8 ⊢ ((𝐷 ∈ Q ∧ 𝐸 ∈ Q) → (𝐷 ·Q 𝐸) ∈ Q)
16	5, 3, 15	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·Q 𝐸) ∈ Q)
17		ltmnqg 7521	. . . . . . 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 1250	. . . . . 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 7688	. . . . 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 1249	. . . 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 735	. 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 7503	. . . . . . 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 7496	. . . . . . . . . 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 7521	. . . . . . . . 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 1250	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·Q 𝑈) <Q 𝑅 ↔ (𝐷 ·Q (𝑇 ·Q 𝑈)) <Q (𝐷 ·Q 𝑅)))
43	34, 5, 6, 8, 10	caov12d 6135	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ·Q (𝐷 ·Q 𝑈)) = (𝐷 ·Q (𝑇 ·Q 𝑈)))
44	43	breq1d 4057	. . . . . . . 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 4071	. . . . 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 7689	. . . . 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 1249	. . . 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 736	. 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 7496	. . . . . . 7 ⊢ ((𝐷 ∈ Q ∧ 𝑈 ∈ Q) → (𝐷 ·Q 𝑈) ∈ Q)
60	4, 59	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐷 ·Q 𝑈) ∈ Q)
61		ltmnqg 7521	. . . . . 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 1250	. . . . 5 ⊢ (𝜑 → (𝐸 <Q 𝑇 ↔ ((𝐷 ·Q 𝑈) ·Q 𝐸) <Q ((𝐷 ·Q 𝑈) ·Q 𝑇)))
63		mulcomnqg 7503	. . . . . . 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 7503	. . . . . . 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 4060	. . . . 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 7595	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
71		prloc 7611	. . . 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 800	1 ⊢ (𝜑 → (𝑄 ∈ (1st ‘(𝐴 ·P 𝐵)) ∨ 𝑅 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ⟨cop 3637   class class class wbr 4047  ‘cfv 5276  (class class class)co 5951  1^st c1st 6231  2^nd c2nd 6232  Qcnq 7400   ·_Q cmq 7403   <_Q cltq 7405  Pcnp 7411   ·_P cmp 7414

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-eprel 4340  df-id 4344  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-1o 6509  df-oadd 6513  df-omul 6514  df-er 6627  df-ec 6629  df-qs 6633  df-ni 7424  df-mi 7426  df-lti 7427  df-mpq 7465  df-enq 7467  df-nqqs 7468  df-mqqs 7470  df-1nqqs 7471  df-rq 7472  df-ltnqqs 7473  df-inp 7586  df-imp 7589

This theorem is referenced by:  mullocpr  7691