Theorem mulclprlem 10633

Description: Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
mulclprlem	⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) → 𝑥 ∈ (𝐴 ·P 𝐵)))

Distinct variable groups:   𝑥,𝑔,ℎ   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑔,ℎ)   𝐵(𝑔,ℎ)

Proof of Theorem mulclprlem

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

Step	Hyp	Ref	Expression
1		elprnq 10605	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
2		elprnq 10605	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
3		recclnq 10580	. . . . . . . . 9 ⊢ (ℎ ∈ Q → (*Q‘ℎ) ∈ Q)
4	3	adantl 485	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (*Q‘ℎ) ∈ Q)
5		vex 3412	. . . . . . . . 9 ⊢ 𝑥 ∈ V
6		ovex 7246	. . . . . . . . 9 ⊢ (𝑔 ·Q ℎ) ∈ V
7		ltmnq 10586	. . . . . . . . 9 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8		fvex 6730	. . . . . . . . 9 ⊢ (*Q‘ℎ) ∈ V
9		mulcomnq 10567	. . . . . . . . 9 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
10	5, 6, 7, 8, 9	caovord2 7420	. . . . . . . 8 ⊢ ((*Q‘ℎ) ∈ Q → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ))))
11	4, 10	syl 17	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ))))
12		mulassnq 10573	. . . . . . . . . 10 ⊢ ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q (ℎ ·Q (*Q‘ℎ)))
13		recidnq 10579	. . . . . . . . . . 11 ⊢ (ℎ ∈ Q → (ℎ ·Q (*Q‘ℎ)) = 1Q)
14	13	oveq2d 7229	. . . . . . . . . 10 ⊢ (ℎ ∈ Q → (𝑔 ·Q (ℎ ·Q (*Q‘ℎ))) = (𝑔 ·Q 1Q))
15	12, 14	eqtrid 2789	. . . . . . . . 9 ⊢ (ℎ ∈ Q → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q 1Q))
16		mulidnq 10577	. . . . . . . . 9 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔)
17	15, 16	sylan9eqr 2800	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = 𝑔)
18	17	breq2d 5065	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑥 ·Q (*Q‘ℎ)) <Q ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
19	11, 18	bitrd 282	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
20	1, 2, 19	syl2an 599	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
21		prcdnq 10607	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
22	21	adantr 484	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
23	20, 22	sylbid 243	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
24		df-mp 10598	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25		mulclnq 10561	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
26	24, 25	genpprecl 10615	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
27	26	exp4b 434	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → (ℎ ∈ 𝐵 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))))
28	27	com34 91	. . . . . 6 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (ℎ ∈ 𝐵 → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))))
29	28	imp32 422	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
30	29	adantlr 715	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
31	23, 30	syld 47	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
32	31	adantr 484	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
33	2	adantl 485	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ℎ ∈ Q)
34		mulassnq 10573	. . . . . 6 ⊢ ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ))
35		mulcomnq 10567	. . . . . . . 8 ⊢ ((*Q‘ℎ) ·Q ℎ) = (ℎ ·Q (*Q‘ℎ))
36	35, 13	eqtrid 2789	. . . . . . 7 ⊢ (ℎ ∈ Q → ((*Q‘ℎ) ·Q ℎ) = 1Q)
37	36	oveq2d 7229	. . . . . 6 ⊢ (ℎ ∈ Q → (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ)) = (𝑥 ·Q 1Q))
38	34, 37	eqtrid 2789	. . . . 5 ⊢ (ℎ ∈ Q → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q 1Q))
39		mulidnq 10577	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
40	38, 39	sylan9eq 2798	. . . 4 ⊢ ((ℎ ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = 𝑥)
41	40	eleq1d 2822	. . 3 ⊢ ((ℎ ∈ Q ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
42	33, 41	sylan 583	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
43	32, 42	sylibd 242	1 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) → 𝑥 ∈ (𝐴 ·P 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110   class class class wbr 5053  ‘cfv 6380  (class class class)co 7213  Qcnq 10466  1_Qc1q 10467   ·_Q cmq 10470  *_Qcrq 10471   <_Q cltq 10472  Pcnp 10473   ·_P cmp 10476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-oadd 8206  df-omul 8207  df-er 8391  df-ni 10486  df-mi 10488  df-lti 10489  df-mpq 10523  df-ltpq 10524  df-enq 10525  df-nq 10526  df-erq 10527  df-mq 10529  df-1nq 10530  df-rq 10531  df-ltnq 10532  df-np 10595  df-mp 10598

This theorem is referenced by:  mulclpr  10634