Theorem addclprlem2 10439

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 10438	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
2	1	adantlr 713	. . . 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 10438	. . . . . 6 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (ℎ +Q 𝑔) → ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵))
4		addcomnq 10373	. . . . . . 7 ⊢ (𝑔 +Q ℎ) = (ℎ +Q 𝑔)
5	4	breq2i 5074	. . . . . 6 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ 𝑥 <Q (ℎ +Q 𝑔))
6	4	fveq2i 6673	. . . . . . . . 9 ⊢ (*Q‘(𝑔 +Q ℎ)) = (*Q‘(ℎ +Q 𝑔))
7	6	oveq2i 7167	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) = (𝑥 ·Q (*Q‘(ℎ +Q 𝑔)))
8	7	oveq1i 7166	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) = ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ)
9	8	eleq1i 2903	. . . . . 6 ⊢ (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵)
10	3, 5, 9	3imtr4g 298	. . . . 5 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵))
11	10	adantll 712	. . . 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 515	. . 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 485	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)))
14		simpl 485	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝐴 ∈ P)
15		simpl 485	. . . . 5 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → 𝐵 ∈ P)
16	14, 15	anim12i 614	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝐴 ∈ P ∧ 𝐵 ∈ P))
17		df-plp 10405	. . . . 5 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)})
18		addclnq 10367	. . . . 5 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q)
19	17, 18	genpprecl 10423	. . . 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 10383	. . . . 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 10381	. . . . 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 2846	. . . 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 10375	. . . . . . 7 ⊢ ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = ((𝑔 +Q ℎ) ·Q (*Q‘(𝑔 +Q ℎ)))
26		elprnq 10413	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
27		elprnq 10413	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
28	26, 27	anim12i 614	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑔 ∈ Q ∧ ℎ ∈ Q))
29		addclnq 10367	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
30		recidnq 10387	. . . . . . . 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 2868	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = 1Q)
33	32	oveq2d 7172	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ))) = (𝑥 ·Q 1Q))
34		mulidnq 10385	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
35	33, 34	sylan9eq 2876	. . . 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 2868	. . 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 2897	. 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 241	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 398   = wceq 1537   ∈ wcel 2114   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Qcnq 10274  1_Qc1q 10275   +_Q cplq 10277   ·_Q cmq 10278  *_Qcrq 10279   <_Q cltq 10280  Pcnp 10281   +_P cpp 10283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3496  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 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-omul 8107  df-er 8289  df-ni 10294  df-pli 10295  df-mi 10296  df-lti 10297  df-plpq 10330  df-mpq 10331  df-ltpq 10332  df-enq 10333  df-nq 10334  df-erq 10335  df-plq 10336  df-mq 10337  df-1nq 10338  df-rq 10339  df-ltnq 10340  df-np 10403  df-plp 10405

This theorem is referenced by:  addclpr  10440