Theorem addclprlem1 11131

Description: Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
addclprlem1	⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))

Proof of Theorem addclprlem1

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

Step	Hyp	Ref	Expression
1		elprnq 11106	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
2		ltrnq 11094	. . . . 5 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ (*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥))
3		ltmnq 11087	. . . . . 6 ⊢ (𝑥 ∈ Q → ((*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) <Q (𝑥 ·Q (*Q‘𝑥))))
4		ovex 7479	. . . . . . 7 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ∈ V
5		ovex 7479	. . . . . . 7 ⊢ (𝑥 ·Q (*Q‘𝑥)) ∈ V
6		ltmnq 11087	. . . . . . 7 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7		vex 3458	. . . . . . 7 ⊢ 𝑔 ∈ V
8		mulcomnq 11068	. . . . . . 7 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
9	4, 5, 6, 7, 8	caovord2 7660	. . . . . 6 ⊢ (𝑔 ∈ Q → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) <Q (𝑥 ·Q (*Q‘𝑥)) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔)))
10	3, 9	sylan9bbr 520	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → ((*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔)))
11	2, 10	bitrid 286	. . . 4 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔)))
12		recidnq 11080	. . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
13	12	oveq1d 7461	. . . . . 6 ⊢ (𝑥 ∈ Q → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14		mulcomnq 11068	. . . . . . 7 ⊢ (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15		mulidnq 11078	. . . . . . 7 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔)
16	14, 15	eqtrid 2810	. . . . . 6 ⊢ (𝑔 ∈ Q → (1Q ·Q 𝑔) = 𝑔)
17	13, 16	sylan9eqr 2820	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = 𝑔)
18	17	breq2d 5118	. . . 4 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔))
19	11, 18	bitrd 282	. . 3 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔))
20	1, 19	sylan 592	. 2 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔))
21		prcdnq 11108	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
22	21	adantr 486	. 2 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔 → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
23	20, 22	sylbid 243	1 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))

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 cplq 10970   ·_Q cmq 10971  *_Qcrq 10972   <_Q cltq 10973  Pcnp 10974

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-pr 5396  ax-un 7778

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

This theorem is used by:  addclprlem2  11132