Theorem addclprlem1 10432

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 10407	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
2		ltrnq 10395	. . . . 5 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ (*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥))
3		ltmnq 10388	. . . . . 6 ⊢ (𝑥 ∈ Q → ((*Q‘(𝑔 +Q ℎ)) <Q (*Q‘𝑥) ↔ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) <Q (𝑥 ·Q (*Q‘𝑥))))
4		ovex 7183	. . . . . . 7 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ∈ V
5		ovex 7183	. . . . . . 7 ⊢ (𝑥 ·Q (*Q‘𝑥)) ∈ V
6		ltmnq 10388	. . . . . . 7 ⊢ (𝑤 ∈ Q → (𝑦 <Q 𝑧 ↔ (𝑤 ·Q 𝑦) <Q (𝑤 ·Q 𝑧)))
7		vex 3498	. . . . . . 7 ⊢ 𝑔 ∈ V
8		mulcomnq 10369	. . . . . . 7 ⊢ (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
9	4, 5, 6, 7, 8	caovord2 7354	. . . . . 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 513	. . . . 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	syl5bb 285	. . . 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 10381	. . . . . . 7 ⊢ (𝑥 ∈ Q → (𝑥 ·Q (*Q‘𝑥)) = 1Q)
13	12	oveq1d 7165	. . . . . 6 ⊢ (𝑥 ∈ Q → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = (1Q ·Q 𝑔))
14		mulcomnq 10369	. . . . . . 7 ⊢ (1Q ·Q 𝑔) = (𝑔 ·Q 1Q)
15		mulidnq 10379	. . . . . . 7 ⊢ (𝑔 ∈ Q → (𝑔 ·Q 1Q) = 𝑔)
16	14, 15	syl5eq 2868	. . . . . 6 ⊢ (𝑔 ∈ Q → (1Q ·Q 𝑔) = 𝑔)
17	13, 16	sylan9eqr 2878	. . . . 5 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → ((𝑥 ·Q (*Q‘𝑥)) ·Q 𝑔) = 𝑔)
18	17	breq2d 5071	. . . 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 281	. . 3 ⊢ ((𝑔 ∈ Q ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔))
20	1, 19	sylan 582	. 2 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) ↔ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) <Q 𝑔))
21		prcdnq 10409	. . 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 483	. 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 242	1 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110   class class class wbr 5059  ‘cfv 6350  (class class class)co 7150  Qcnq 10268  1_Qc1q 10269   +_Q cplq 10271   ·_Q cmq 10272  *_Qcrq 10273   <_Q cltq 10274  Pcnp 10275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-omul 8101  df-er 8283  df-ni 10288  df-mi 10290  df-lti 10291  df-mpq 10325  df-ltpq 10326  df-enq 10327  df-nq 10328  df-erq 10329  df-mq 10331  df-1nq 10332  df-rq 10333  df-ltnq 10334  df-np 10397

This theorem is referenced by:  addclprlem2  10433