Theorem prmuloclemcalc 7896

Description: Calculations for prmuloc 7897. (Contributed by Jim Kingdon, 9-Dec-2019.)

Hypotheses

Ref	Expression
prmuloclemcalc.ru	⊢ (𝜑 → 𝑅 <Q 𝑈)
prmuloclemcalc.udp	⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
prmuloclemcalc.axb	⊢ (𝜑 → (𝐴 +Q 𝑋) = 𝐵)
prmuloclemcalc.pbrx	⊢ (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))
prmuloclemcalc.a	⊢ (𝜑 → 𝐴 ∈ Q)
prmuloclemcalc.b	⊢ (𝜑 → 𝐵 ∈ Q)
prmuloclemcalc.d	⊢ (𝜑 → 𝐷 ∈ Q)
prmuloclemcalc.p	⊢ (𝜑 → 𝑃 ∈ Q)
prmuloclemcalc.x	⊢ (𝜑 → 𝑋 ∈ Q)

Assertion

Ref	Expression
prmuloclemcalc	⊢ (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))

Proof of Theorem prmuloclemcalc

Step	Hyp	Ref	Expression
1		prmuloclemcalc.axb	. . . . . . 7 ⊢ (𝜑 → (𝐴 +Q 𝑋) = 𝐵)
2	1	oveq2d 6074	. . . . . 6 ⊢ (𝜑 → (𝑈 ·Q (𝐴 +Q 𝑋)) = (𝑈 ·Q 𝐵))
3		prmuloclemcalc.ru	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 <Q 𝑈)
4		ltrelnq 7696	. . . . . . . . . 10 ⊢ <Q ⊆ (Q × Q)
5	4	brel 4807	. . . . . . . . 9 ⊢ (𝑅 <Q 𝑈 → (𝑅 ∈ Q ∧ 𝑈 ∈ Q))
6	3, 5	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ∈ Q ∧ 𝑈 ∈ Q))
7	6	simprd 114	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ Q)
8		prmuloclemcalc.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ Q)
9		prmuloclemcalc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ Q)
10		distrnqg 7718	. . . . . . 7 ⊢ ((𝑈 ∈ Q ∧ 𝐴 ∈ Q ∧ 𝑋 ∈ Q) → (𝑈 ·Q (𝐴 +Q 𝑋)) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
11	7, 8, 9, 10	syl3anc 1274	. . . . . 6 ⊢ (𝜑 → (𝑈 ·Q (𝐴 +Q 𝑋)) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
12	2, 11	eqtr3d 2269	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝐵) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
13		prmuloclemcalc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Q)
14		mulcomnqg 7714	. . . . . . 7 ⊢ ((𝐵 ∈ Q ∧ 𝑈 ∈ Q) → (𝐵 ·Q 𝑈) = (𝑈 ·Q 𝐵))
15	13, 7, 14	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝐵 ·Q 𝑈) = (𝑈 ·Q 𝐵))
16		prmuloclemcalc.udp	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
17		ltmnqi 7734	. . . . . . . . . 10 ⊢ ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝐵 ∈ Q) → (𝐵 ·Q 𝑈) <Q (𝐵 ·Q (𝐷 +Q 𝑃)))
18	16, 13, 17	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q (𝐵 ·Q (𝐷 +Q 𝑃)))
19		prmuloclemcalc.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ Q)
20		prmuloclemcalc.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ Q)
21		distrnqg 7718	. . . . . . . . . 10 ⊢ ((𝐵 ∈ Q ∧ 𝐷 ∈ Q ∧ 𝑃 ∈ Q) → (𝐵 ·Q (𝐷 +Q 𝑃)) = ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
22	13, 19, 20, 21	syl3anc 1274	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q (𝐷 +Q 𝑃)) = ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
23	18, 22	breqtrd 4140	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
24		mulcomnqg 7714	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Q ∧ 𝐵 ∈ Q) → (𝑃 ·Q 𝐵) = (𝐵 ·Q 𝑃))
25	20, 13, 24	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ·Q 𝐵) = (𝐵 ·Q 𝑃))
26		prmuloclemcalc.pbrx	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))
27	25, 26	eqbrtrrd 4138	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q 𝑃) <Q (𝑅 ·Q 𝑋))
28		mulclnq 7707	. . . . . . . . . 10 ⊢ ((𝐵 ∈ Q ∧ 𝐷 ∈ Q) → (𝐵 ·Q 𝐷) ∈ Q)
29	13, 19, 28	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q 𝐷) ∈ Q)
30		ltanqi 7733	. . . . . . . . 9 ⊢ (((𝐵 ·Q 𝑃) <Q (𝑅 ·Q 𝑋) ∧ (𝐵 ·Q 𝐷) ∈ Q) → ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
31	27, 29, 30	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
32		ltsonq 7729	. . . . . . . . 9 ⊢ <Q Or Q
33	32, 4	sotri 5163	. . . . . . . 8 ⊢ (((𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) ∧ ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋))) → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
34	23, 31, 33	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
35		ltmnqi 7734	. . . . . . . . . 10 ⊢ ((𝑅 <Q 𝑈 ∧ 𝑋 ∈ Q) → (𝑋 ·Q 𝑅) <Q (𝑋 ·Q 𝑈))
36	3, 9, 35	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·Q 𝑅) <Q (𝑋 ·Q 𝑈))
37	6	simpld 112	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Q)
38		mulcomnqg 7714	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Q ∧ 𝑅 ∈ Q) → (𝑋 ·Q 𝑅) = (𝑅 ·Q 𝑋))
39	9, 37, 38	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·Q 𝑅) = (𝑅 ·Q 𝑋))
40		mulcomnqg 7714	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Q ∧ 𝑈 ∈ Q) → (𝑋 ·Q 𝑈) = (𝑈 ·Q 𝑋))
41	9, 7, 40	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·Q 𝑈) = (𝑈 ·Q 𝑋))
42	36, 39, 41	3brtr3d 4145	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ·Q 𝑋) <Q (𝑈 ·Q 𝑋))
43		ltanqi 7733	. . . . . . . 8 ⊢ (((𝑅 ·Q 𝑋) <Q (𝑈 ·Q 𝑋) ∧ (𝐵 ·Q 𝐷) ∈ Q) → ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
44	42, 29, 43	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
45	32, 4	sotri 5163	. . . . . . 7 ⊢ (((𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) ∧ ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋))) → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
46	34, 44, 45	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
47	15, 46	eqbrtrrd 4138	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝐵) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
48	12, 47	eqbrtrrd 4138	. . . 4 ⊢ (𝜑 → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
49		mulclnq 7707	. . . . . 6 ⊢ ((𝑈 ∈ Q ∧ 𝐴 ∈ Q) → (𝑈 ·Q 𝐴) ∈ Q)
50	7, 8, 49	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝐴) ∈ Q)
51		mulclnq 7707	. . . . . 6 ⊢ ((𝑈 ∈ Q ∧ 𝑋 ∈ Q) → (𝑈 ·Q 𝑋) ∈ Q)
52	7, 9, 51	syl2anc 411	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝑋) ∈ Q)
53		addcomnqg 7712	. . . . 5 ⊢ (((𝑈 ·Q 𝐴) ∈ Q ∧ (𝑈 ·Q 𝑋) ∈ Q) → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)))
54	50, 52, 53	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)))
55		addcomnqg 7712	. . . . 5 ⊢ (((𝐵 ·Q 𝐷) ∈ Q ∧ (𝑈 ·Q 𝑋) ∈ Q) → ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
56	29, 52, 55	syl2anc 411	. . . 4 ⊢ (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
57	48, 54, 56	3brtr3d 4145	. . 3 ⊢ (𝜑 → ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
58		ltanqg 7731	. . . 4 ⊢ (((𝑈 ·Q 𝐴) ∈ Q ∧ (𝐵 ·Q 𝐷) ∈ Q ∧ (𝑈 ·Q 𝑋) ∈ Q) → ((𝑈 ·Q 𝐴) <Q (𝐵 ·Q 𝐷) ↔ ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷))))
59	50, 29, 52, 58	syl3anc 1274	. . 3 ⊢ (𝜑 → ((𝑈 ·Q 𝐴) <Q (𝐵 ·Q 𝐷) ↔ ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷))))
60	57, 59	mpbird 167	. 2 ⊢ (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐵 ·Q 𝐷))
61		mulcomnqg 7714	. . 3 ⊢ ((𝐵 ∈ Q ∧ 𝐷 ∈ Q) → (𝐵 ·Q 𝐷) = (𝐷 ·Q 𝐵))
62	13, 19, 61	syl2anc 411	. 2 ⊢ (𝜑 → (𝐵 ·Q 𝐷) = (𝐷 ·Q 𝐵))
63	60, 62	breqtrd 4140	1 ⊢ (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   class class class wbr 4114  (class class class)co 6058  Qcnq 7611   +_Q cplq 7613   ·_Q cmq 7614   <_Q cltq 7616

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-ltnqqs 7684

This theorem is referenced by:  prmuloc  7897