Theorem prmuloclemcalc 7926

Description: Calculations for prmuloc 7927. (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 6095	. . . . . 6 ⊢ (𝜑 → (𝑈 ·Q (𝐴 +Q 𝑋)) = (𝑈 ·Q 𝐵))
3		prmuloclemcalc.ru	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 <Q 𝑈)
4		ltrelnq 7726	. . . . . . . . . 10 ⊢ <Q ⊆ (Q × Q)
5	4	brel 4825	. . . . . . . . 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 7748	. . . . . . 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 1278	. . . . . 6 ⊢ (𝜑 → (𝑈 ·Q (𝐴 +Q 𝑋)) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
12	2, 11	eqtr3d 2273	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝐵) = ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)))
13		prmuloclemcalc.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ Q)
14		mulcomnqg 7744	. . . . . . 7 ⊢ ((𝐵 ∈ Q ∧ 𝑈 ∈ Q) → (𝐵 ·Q 𝑈) = (𝑈 ·Q 𝐵))
15	13, 7, 14	syl2anc 415	. . . . . 6 ⊢ (𝜑 → (𝐵 ·Q 𝑈) = (𝑈 ·Q 𝐵))
16		prmuloclemcalc.udp	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 <Q (𝐷 +Q 𝑃))
17		ltmnqi 7764	. . . . . . . . . 10 ⊢ ((𝑈 <Q (𝐷 +Q 𝑃) ∧ 𝐵 ∈ Q) → (𝐵 ·Q 𝑈) <Q (𝐵 ·Q (𝐷 +Q 𝑃)))
18	16, 13, 17	syl2anc 415	. . . . . . . . 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 7748	. . . . . . . . . 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 1278	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q (𝐷 +Q 𝑃)) = ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
23	18, 22	breqtrd 4154	. . . . . . . 8 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)))
24		mulcomnqg 7744	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Q ∧ 𝐵 ∈ Q) → (𝑃 ·Q 𝐵) = (𝐵 ·Q 𝑃))
25	20, 13, 24	syl2anc 415	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ·Q 𝐵) = (𝐵 ·Q 𝑃))
26		prmuloclemcalc.pbrx	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ·Q 𝐵) <Q (𝑅 ·Q 𝑋))
27	25, 26	eqbrtrrd 4152	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q 𝑃) <Q (𝑅 ·Q 𝑋))
28		mulclnq 7737	. . . . . . . . . 10 ⊢ ((𝐵 ∈ Q ∧ 𝐷 ∈ Q) → (𝐵 ·Q 𝐷) ∈ Q)
29	13, 19, 28	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ·Q 𝐷) ∈ Q)
30		ltanqi 7763	. . . . . . . . 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 415	. . . . . . . 8 ⊢ (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝐵 ·Q 𝑃)) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
32		ltsonq 7759	. . . . . . . . 9 ⊢ <Q Or Q
33	32, 4	sotri 5181	. . . . . . . 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 415	. . . . . . 7 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)))
35		ltmnqi 7764	. . . . . . . . . 10 ⊢ ((𝑅 <Q 𝑈 ∧ 𝑋 ∈ Q) → (𝑋 ·Q 𝑅) <Q (𝑋 ·Q 𝑈))
36	3, 9, 35	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·Q 𝑅) <Q (𝑋 ·Q 𝑈))
37	6	simpld 112	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Q)
38		mulcomnqg 7744	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Q ∧ 𝑅 ∈ Q) → (𝑋 ·Q 𝑅) = (𝑅 ·Q 𝑋))
39	9, 37, 38	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·Q 𝑅) = (𝑅 ·Q 𝑋))
40		mulcomnqg 7744	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Q ∧ 𝑈 ∈ Q) → (𝑋 ·Q 𝑈) = (𝑈 ·Q 𝑋))
41	9, 7, 40	syl2anc 415	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ·Q 𝑈) = (𝑈 ·Q 𝑋))
42	36, 39, 41	3brtr3d 4159	. . . . . . . 8 ⊢ (𝜑 → (𝑅 ·Q 𝑋) <Q (𝑈 ·Q 𝑋))
43		ltanqi 7763	. . . . . . . 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 415	. . . . . . 7 ⊢ (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝑅 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
45	32, 4	sotri 5181	. . . . . . 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 415	. . . . . 6 ⊢ (𝜑 → (𝐵 ·Q 𝑈) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
47	15, 46	eqbrtrrd 4152	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝐵) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
48	12, 47	eqbrtrrd 4152	. . . 4 ⊢ (𝜑 → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) <Q ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)))
49		mulclnq 7737	. . . . . 6 ⊢ ((𝑈 ∈ Q ∧ 𝐴 ∈ Q) → (𝑈 ·Q 𝐴) ∈ Q)
50	7, 8, 49	syl2anc 415	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝐴) ∈ Q)
51		mulclnq 7737	. . . . . 6 ⊢ ((𝑈 ∈ Q ∧ 𝑋 ∈ Q) → (𝑈 ·Q 𝑋) ∈ Q)
52	7, 9, 51	syl2anc 415	. . . . 5 ⊢ (𝜑 → (𝑈 ·Q 𝑋) ∈ Q)
53		addcomnqg 7742	. . . . 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 415	. . . 4 ⊢ (𝜑 → ((𝑈 ·Q 𝐴) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)))
55		addcomnqg 7742	. . . . 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 415	. . . 4 ⊢ (𝜑 → ((𝐵 ·Q 𝐷) +Q (𝑈 ·Q 𝑋)) = ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
57	48, 54, 56	3brtr3d 4159	. . 3 ⊢ (𝜑 → ((𝑈 ·Q 𝑋) +Q (𝑈 ·Q 𝐴)) <Q ((𝑈 ·Q 𝑋) +Q (𝐵 ·Q 𝐷)))
58		ltanqg 7761	. . . 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 1278	. . 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 7744	. . 3 ⊢ ((𝐵 ∈ Q ∧ 𝐷 ∈ Q) → (𝐵 ·Q 𝐷) = (𝐷 ·Q 𝐵))
62	13, 19, 61	syl2anc 415	. 2 ⊢ (𝜑 → (𝐵 ·Q 𝐷) = (𝐷 ·Q 𝐵))
63	60, 62	breqtrd 4154	1 ⊢ (𝜑 → (𝑈 ·Q 𝐴) <Q (𝐷 ·Q 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   class class class wbr 4128  (class class class)co 6079  Qcnq 7641   +_Q cplq 7643   ·_Q cmq 7644   <_Q cltq 7646

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-po 4439  df-iso 4440  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-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-ltnqqs 7714

This theorem is referenced by:  prmuloc  7927