Theorem mulnqpru 7568

Description: Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)

Assertion

Ref	Expression
mulnqpru	⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Proof of Theorem mulnqpru

Dummy variables 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltmnqg 7400	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
2	1	adantl 277	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q ∧ 𝑤 ∈ Q)) → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
3		prop 7474	. . . . . . . . 9 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
4		elprnqu 7481	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
5	3, 4	sylan 283	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → 𝐺 ∈ Q)
6	5	ad2antrr 488	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐺 ∈ Q)
7		prop 7474	. . . . . . . . 9 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
8		elprnqu 7481	. . . . . . . . 9 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
9	7, 8	sylan 283	. . . . . . . 8 ⊢ ((𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵)) → 𝐻 ∈ Q)
10	9	ad2antlr 489	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝐻 ∈ Q)
11		mulclnq 7375	. . . . . . 7 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
12	6, 10, 11	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 𝐻) ∈ Q)
13		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → 𝑋 ∈ Q)
14		recclnq 7391	. . . . . . 7 ⊢ (𝐻 ∈ Q → (*Q‘𝐻) ∈ Q)
15	10, 14	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (*Q‘𝐻) ∈ Q)
16		mulcomnqg 7382	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
17	16	adantl 277	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) ∧ (𝑦 ∈ Q ∧ 𝑧 ∈ Q)) → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
18	2, 12, 13, 15, 17	caovord2d 6044	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) <Q (𝑋 ·Q (*Q‘𝐻))))
19		mulassnqg 7383	. . . . . . . 8 ⊢ ((𝐺 ∈ Q ∧ 𝐻 ∈ Q ∧ (*Q‘𝐻) ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))))
20	6, 10, 15, 19	syl3anc 1238	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))))
21		recidnq 7392	. . . . . . . . 9 ⊢ (𝐻 ∈ Q → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
22	21	oveq2d 5891	. . . . . . . 8 ⊢ (𝐻 ∈ Q → (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))) = (𝐺 ·Q 1Q))
23	10, 22	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q (𝐻 ·Q (*Q‘𝐻))) = (𝐺 ·Q 1Q))
24		mulidnq 7388	. . . . . . . 8 ⊢ (𝐺 ∈ Q → (𝐺 ·Q 1Q) = 𝐺)
25	6, 24	syl 14	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 ·Q 1Q) = 𝐺)
26	20, 23, 25	3eqtrd 2214	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) = 𝐺)
27	26	breq1d 4014	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝐺 ·Q 𝐻) ·Q (*Q‘𝐻)) <Q (𝑋 ·Q (*Q‘𝐻)) ↔ 𝐺 <Q (𝑋 ·Q (*Q‘𝐻))))
28	18, 27	bitrd 188	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 ↔ 𝐺 <Q (𝑋 ·Q (*Q‘𝐻))))
29		prcunqu 7484	. . . . . 6 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
30	3, 29	sylan 283	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
31	30	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐺 <Q (𝑋 ·Q (*Q‘𝐻)) → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
32	28, 31	sylbid 150	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → (𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴)))
33		df-imp 7468	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦 ·Q 𝑧))}⟩)
34		mulclnq 7375	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
35	33, 34	genppreclu 7514	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) ∧ 𝐻 ∈ (2nd ‘𝐵)) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
36	35	exp4b 367	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → (𝐻 ∈ (2nd ‘𝐵) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))))
37	36	com34 83	. . . . . 6 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝐻 ∈ (2nd ‘𝐵) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))))
38	37	imp32 257	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
39	38	adantlr 477	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
40	39	adantr 276	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ∈ (2nd ‘𝐴) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
41	32, 40	syld 45	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵))))
42		mulassnqg 7383	. . . . 5 ⊢ ((𝑋 ∈ Q ∧ (*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)))
43	13, 15, 10, 42	syl3anc 1238	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)))
44		mulcomnqg 7382	. . . . . . 7 ⊢ (((*Q‘𝐻) ∈ Q ∧ 𝐻 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
45	15, 10, 44	syl2anc 411	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = (𝐻 ·Q (*Q‘𝐻)))
46	10, 21	syl 14	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝐻 ·Q (*Q‘𝐻)) = 1Q)
47	45, 46	eqtrd 2210	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((*Q‘𝐻) ·Q 𝐻) = 1Q)
48	47	oveq2d 5891	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q ((*Q‘𝐻) ·Q 𝐻)) = (𝑋 ·Q 1Q))
49		mulidnq 7388	. . . . 5 ⊢ (𝑋 ∈ Q → (𝑋 ·Q 1Q) = 𝑋)
50	49	adantl 277	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (𝑋 ·Q 1Q) = 𝑋)
51	43, 48, 50	3eqtrd 2214	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) = 𝑋)
52	51	eleq1d 2246	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → (((𝑋 ·Q (*Q‘𝐻)) ·Q 𝐻) ∈ (2nd ‘(𝐴 ·P 𝐵)) ↔ 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))
53	41, 52	sylibd 149	1 ⊢ ((((𝐴 ∈ P ∧ 𝐺 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝐻 ∈ (2nd ‘𝐵))) ∧ 𝑋 ∈ Q) → ((𝐺 ·Q 𝐻) <Q 𝑋 → 𝑋 ∈ (2nd ‘(𝐴 ·P 𝐵))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ⟨cop 3596   class class class wbr 4004  ‘cfv 5217  (class class class)co 5875  1^st c1st 6139  2^nd c2nd 6140  Qcnq 7279  1_Qc1q 7280   ·_Q cmq 7282  *_Qcrq 7283   <_Q cltq 7284  Pcnp 7290   ·_P cmp 7293

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-eprel 4290  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-1o 6417  df-oadd 6421  df-omul 6422  df-er 6535  df-ec 6537  df-qs 6541  df-ni 7303  df-mi 7305  df-lti 7306  df-mpq 7344  df-enq 7346  df-nqqs 7347  df-mqqs 7349  df-1nqqs 7350  df-rq 7351  df-ltnqqs 7352  df-inp 7465  df-imp 7468

This theorem is referenced by:  mullocprlem  7569  mulclpr  7571