Theorem mulclprlem 11134

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 11106	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
2		elprnq 11106	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
3		recclnq 11081	. . . . . . . . 9 ⊢ (ℎ ∈ Q → (*Q‘ℎ) ∈ Q)
4	3	adantl 487	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (*Q‘ℎ) ∈ Q)
5		vex 3458	. . . . . . . . 9 ⊢ 𝑥 ∈ V
6		ovex 7479	. . . . . . . . 9 ⊢ (𝑔 ·Q ℎ) ∈ V
7		ltmnq 11087	. . . . . . . . 9 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8		fvex 6924	. . . . . . . . 9 ⊢ (*Q‘ℎ) ∈ V
9		mulcomnq 11068	. . . . . . . . 9 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
10	5, 6, 7, 8, 9	caovord2 7660	. . . . . . . 8 ⊢ ((*Q‘ℎ) ∈ Q → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ))))
11	4, 10	syl 18	. . . . . . 7 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ))))
12		mulassnq 11074	. . . . . . . . . 10 ⊢ ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q (ℎ ·Q (*Q‘ℎ)))
13		recidnq 11080	. . . . . . . . . . 11 ⊢ (ℎ ∈ Q → (ℎ ·Q (*Q‘ℎ)) = 1Q)
14	13	oveq2d 7462	. . . . . . . . . 10 ⊢ (ℎ ∈ Q → (𝑔 ·Q (ℎ ·Q (*Q‘ℎ))) = (𝑔 ·Q 1Q))
15	12, 14	eqtrid 2810	. . . . . . . . 9 ⊢ (ℎ ∈ Q → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q 1Q))
16		mulidnq 11078	. . . . . . . . 9 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔)
17	15, 16	sylan9eqr 2820	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = 𝑔)
18	17	breq2d 5118	. . . . . . 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 608	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
21		prcdnq 11108	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
22	21	adantr 486	. . . . 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 11099	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25		mulclnq 11062	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
26	24, 25	genpprecl 11116	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
27	26	exp4b 436	. . . . . . 7 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → (ℎ ∈ 𝐵 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))))
28	27	com34 92	. . . . . 6 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (ℎ ∈ 𝐵 → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))))
29	28	imp32 424	. . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
30	29	adantlr 728	. . . 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 48	. . 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 486	. 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 487	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ℎ ∈ Q)
34		mulassnq 11074	. . . . . 6 ⊢ ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ))
35		mulcomnq 11068	. . . . . . . 8 ⊢ ((*Q‘ℎ) ·Q ℎ) = (ℎ ·Q (*Q‘ℎ))
36	35, 13	eqtrid 2810	. . . . . . 7 ⊢ (ℎ ∈ Q → ((*Q‘ℎ) ·Q ℎ) = 1Q)
37	36	oveq2d 7462	. . . . . 6 ⊢ (ℎ ∈ Q → (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ)) = (𝑥 ·Q 1Q))
38	34, 37	eqtrid 2810	. . . . 5 ⊢ (ℎ ∈ Q → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q 1Q))
39		mulidnq 11078	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
40	38, 39	sylan9eq 2818	. . . 4 ⊢ ((ℎ ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = 𝑥)
41	40	eleq1d 2849	. . 3 ⊢ ((ℎ ∈ Q ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵) ↔ 𝑥 ∈ (𝐴 ·P 𝐵)))
42	33, 41	sylan 592	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446  Qcnq 10967  1_Qc1q 10968   ·_Q cmq 10971  *_Qcrq 10972   <_Q cltq 10973  Pcnp 10974   ·_P cmp 10977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-inf2 9672

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-oadd 8511  df-omul 8512  df-er 8748  df-ni 10987  df-mi 10989  df-lti 10990  df-mpq 11024  df-ltpq 11025  df-enq 11026  df-nq 11027  df-erq 11028  df-mq 11030  df-1nq 11031  df-rq 11032  df-ltnq 11033  df-np 11096  df-mp 11099

This theorem is used by:  mulclpr  11135