Theorem mulclprlem 10955

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 10927	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
2		elprnq 10927	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
3		recclnq 10902	. . . . . . . . 9 ⊢ (ℎ ∈ Q → (*Q‘ℎ) ∈ Q)
4	3	adantl 482	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (*Q‘ℎ) ∈ Q)
5		vex 3449	. . . . . . . . 9 ⊢ 𝑥 ∈ V
6		ovex 7390	. . . . . . . . 9 ⊢ (𝑔 ·Q ℎ) ∈ V
7		ltmnq 10908	. . . . . . . . 9 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
8		fvex 6855	. . . . . . . . 9 ⊢ (*Q‘ℎ) ∈ V
9		mulcomnq 10889	. . . . . . . . 9 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
10	5, 6, 7, 8, 9	caovord2 7566	. . . . . . . 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 10895	. . . . . . . . . 10 ⊢ ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q (ℎ ·Q (*Q‘ℎ)))
13		recidnq 10901	. . . . . . . . . . 11 ⊢ (ℎ ∈ Q → (ℎ ·Q (*Q‘ℎ)) = 1Q)
14	13	oveq2d 7373	. . . . . . . . . 10 ⊢ (ℎ ∈ Q → (𝑔 ·Q (ℎ ·Q (*Q‘ℎ))) = (𝑔 ·Q 1Q))
15	12, 14	eqtrid 2788	. . . . . . . . 9 ⊢ (ℎ ∈ Q → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = (𝑔 ·Q 1Q))
16		mulidnq 10899	. . . . . . . . 9 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔)
17	15, 16	sylan9eqr 2798	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑔 ·Q ℎ) ·Q (*Q‘ℎ)) = 𝑔)
18	17	breq2d 5117	. . . . . . 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 278	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
20	1, 2, 19	syl2an 596	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) ↔ (𝑥 ·Q (*Q‘ℎ)) <Q 𝑔))
21		prcdnq 10929	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
22	21	adantr 481	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑥 ·Q (*Q‘ℎ)) <Q 𝑔 → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
23	20, 22	sylbid 239	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 <Q (𝑔 ·Q ℎ) → (𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴))
24		df-mp 10920	. . . . . . . . 9 ⊢ ·P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 ·Q 𝑧)})
25		mulclnq 10883	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·Q 𝑧) ∈ Q)
26	24, 25	genpprecl 10937	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑥 ·Q (*Q‘ℎ)) ∈ 𝐴 ∧ ℎ ∈ 𝐵) → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) ∈ (𝐴 ·P 𝐵)))
27	26	exp4b 431	. . . . . . 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 419	. . . . 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 481	. 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 482	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ℎ ∈ Q)
34		mulassnq 10895	. . . . . 6 ⊢ ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ))
35		mulcomnq 10889	. . . . . . . 8 ⊢ ((*Q‘ℎ) ·Q ℎ) = (ℎ ·Q (*Q‘ℎ))
36	35, 13	eqtrid 2788	. . . . . . 7 ⊢ (ℎ ∈ Q → ((*Q‘ℎ) ·Q ℎ) = 1Q)
37	36	oveq2d 7373	. . . . . 6 ⊢ (ℎ ∈ Q → (𝑥 ·Q ((*Q‘ℎ) ·Q ℎ)) = (𝑥 ·Q 1Q))
38	34, 37	eqtrid 2788	. . . . 5 ⊢ (ℎ ∈ Q → ((𝑥 ·Q (*Q‘ℎ)) ·Q ℎ) = (𝑥 ·Q 1Q))
39		mulidnq 10899	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
40	38, 39	sylan9eq 2796	. . . 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 580	. 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 238	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 205   ∧ wa 396   ∈ wcel 2106   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Qcnq 10788  1_Qc1q 10789   ·_Q cmq 10792  *_Qcrq 10793   <_Q cltq 10794  Pcnp 10795   ·_P cmp 10798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-ni 10808  df-mi 10810  df-lti 10811  df-mpq 10845  df-ltpq 10846  df-enq 10847  df-nq 10848  df-erq 10849  df-mq 10851  df-1nq 10852  df-rq 10853  df-ltnq 10854  df-np 10917  df-mp 10920

This theorem is referenced by:  mulclpr  10956