Theorem addclprlem2 10428

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
addclprlem2	⊢ ((((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 addclprlem2

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

Step	Hyp	Ref	Expression
1		addclprlem1 10427	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
2	1	adantlr 714	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
3		addclprlem1 10427	. . . . . 6 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (ℎ +Q 𝑔) → ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵))
4		addcomnq 10362	. . . . . . 7 ⊢ (𝑔 +Q ℎ) = (ℎ +Q 𝑔)
5	4	breq2i 5038	. . . . . 6 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ 𝑥 <Q (ℎ +Q 𝑔))
6	4	fveq2i 6648	. . . . . . . . 9 ⊢ (*Q‘(𝑔 +Q ℎ)) = (*Q‘(ℎ +Q 𝑔))
7	6	oveq2i 7146	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) = (𝑥 ·Q (*Q‘(ℎ +Q 𝑔)))
8	7	oveq1i 7145	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) = ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ)
9	8	eleq1i 2880	. . . . . 6 ⊢ (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵)
10	3, 5, 9	3imtr4g 299	. . . . 5 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵))
11	10	adantll 713	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵))
12	2, 11	jcad 516	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴 ∧ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵)))
13		simpl 486	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)))
14		simpl 486	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝐴 ∈ P)
15		simpl 486	. . . . 5 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → 𝐵 ∈ P)
16	14, 15	anim12i 615	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
17		df-plp 10394	. . . . 5 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)})
18		addclnq 10356	. . . . 5 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q)
19	17, 18	genpprecl 10412	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴 ∧ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ)) ∈ (𝐴 +P 𝐵)))
20	13, 16, 19	3syl 18	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → ((((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴 ∧ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ)) ∈ (𝐴 +P 𝐵)))
21	12, 20	syld 47	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ)) ∈ (𝐴 +P 𝐵)))
22		distrnq 10372	. . . . 5 ⊢ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q (𝑔 +Q ℎ)) = (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ))
23		mulassnq 10370	. . . . 5 ⊢ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q (𝑔 +Q ℎ)) = (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)))
24	22, 23	eqtr3i 2823	. . . 4 ⊢ (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ)) = (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)))
25		mulcomnq 10364	. . . . . . 7 ⊢ ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = ((𝑔 +Q ℎ) ·Q (*Q‘(𝑔 +Q ℎ)))
26		elprnq 10402	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
27		elprnq 10402	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
28	26, 27	anim12i 615	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑔 ∈ Q ∧ ℎ ∈ Q))
29		addclnq 10356	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
30		recidnq 10376	. . . . . . . 8 ⊢ ((𝑔 +Q ℎ) ∈ Q → ((𝑔 +Q ℎ) ·Q (*Q‘(𝑔 +Q ℎ))) = 1Q)
31	28, 29, 30	3syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑔 +Q ℎ) ·Q (*Q‘(𝑔 +Q ℎ))) = 1Q)
32	25, 31	syl5eq 2845	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = 1Q)
33	32	oveq2d 7151	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ))) = (𝑥 ·Q 1Q))
34		mulidnq 10374	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
35	33, 34	sylan9eq 2853	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ))) = 𝑥)
36	24, 35	syl5eq 2845	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ)) = 𝑥)
37	36	eleq1d 2874	. 2 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → ((((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) +Q ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ)) ∈ (𝐴 +P 𝐵) ↔ 𝑥 ∈ (𝐴 +P 𝐵)))
38	21, 37	sylibd 242	1 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → 𝑥 ∈ (𝐴 +P 𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Qcnq 10263  1_Qc1q 10264   +_Q cplq 10266   ·_Q cmq 10267  *_Qcrq 10268   <_Q cltq 10269  Pcnp 10270   +_P cpp 10272

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090  df-er 8272  df-ni 10283  df-pli 10284  df-mi 10285  df-lti 10286  df-plpq 10319  df-mpq 10320  df-ltpq 10321  df-enq 10322  df-nq 10323  df-erq 10324  df-plq 10325  df-mq 10326  df-1nq 10327  df-rq 10328  df-ltnq 10329  df-np 10392  df-plp 10394

This theorem is referenced by:  addclpr  10429