Theorem mulclprlem 10443

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 10415	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
2		elprnq 10415	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
3		recclnq 10390	. . . . . . . . 9 ⊢ (ℎ ∈ Q → (*Q‘ℎ) ∈ Q)
4	3	adantl 484	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (*Q‘ℎ) ∈ Q)
5		vex 3499	. . . . . . . . 9 ⊢ 𝑥 ∈ V
6		ovex 7191	. . . . . . . . 9 ⊢ (𝑔 ·Q ℎ) ∈ V
7		ltmnq 10396	. . . . . . . . 9 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8		fvex 6685	. . . . . . . . 9 ⊢ (*Q‘ℎ) ∈ V
9		mulcomnq 10377	. . . . . . . . 9 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
10	5, 6, 7, 8, 9	caovord2 7362	. . . . . . . 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 10383	. . . . . . . . . 10 ⊢ ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q (ℎ ·Q (*Q‘ℎ)))
13		recidnq 10389	. . . . . . . . . . 11 ⊢ (ℎ ∈ Q → (ℎ ·Q (*Q‘ℎ)) = 1Q)
14	13	oveq2d 7174	. . . . . . . . . 10 ⊢ (ℎ ∈ Q → (𝑔 ·Q (ℎ ·Q (*Q‘ℎ))) = (𝑔 ·Q 1Q))
15	12, 14	syl5eq 2870	. . . . . . . . 9 ⊢ (ℎ ∈ Q → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q 1Q))
16		mulidnq 10387	. . . . . . . . 9 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔)
17	15, 16	sylan9eqr 2880	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = 𝑔)
18	17	breq2d 5080	. . . . . . 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 281	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
20	1, 2, 19	syl2an 597	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
21		prcdnq 10417	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
22	21	adantr 483	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
23	20, 22	sylbid 242	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
24		df-mp 10408	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25		mulclnq 10371	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
26	24, 25	genpprecl 10425	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
27	26	exp4b 433	. . . . . . 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 421	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
30	29	adantlr 713	. . . 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 483	. 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 484	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ℎ ∈ Q)
34		mulassnq 10383	. . . . . 6 ⊢ ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ))
35		mulcomnq 10377	. . . . . . . 8 ⊢ ((*Q‘ℎ) ·Q ℎ) = (ℎ ·Q (*Q‘ℎ))
36	35, 13	syl5eq 2870	. . . . . . 7 ⊢ (ℎ ∈ Q → ((*Q‘ℎ) ·Q ℎ) = 1Q)
37	36	oveq2d 7174	. . . . . 6 ⊢ (ℎ ∈ Q → (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ)) = (𝑥 ·Q 1Q))
38	34, 37	syl5eq 2870	. . . . 5 ⊢ (ℎ ∈ Q → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q 1Q))
39		mulidnq 10387	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
40	38, 39	sylan9eq 2878	. . . 4 ⊢ ((ℎ ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = 𝑥)
41	40	eleq1d 2899	. . 3 ⊢ ((ℎ ∈ Q ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
42	33, 41	sylan 582	. 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 241	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 208   ∧ wa 398   ∈ wcel 2114   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  Qcnq 10276  1_Qc1q 10277   ·_Q cmq 10280  *_Qcrq 10281   <_Q cltq 10282  Pcnp 10283   ·_P cmp 10286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-omul 8109  df-er 8291  df-ni 10296  df-mi 10298  df-lti 10299  df-mpq 10333  df-ltpq 10334  df-enq 10335  df-nq 10336  df-erq 10337  df-mq 10339  df-1nq 10340  df-rq 10341  df-ltnq 10342  df-np 10405  df-mp 10408

This theorem is referenced by:  mulclpr  10444