Theorem addclprlem2 10940

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 10939	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q 𝑔) ∈ 𝐴))
2	1	adantlr 716	. . . 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 10939	. . . . . 6 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (ℎ +Q 𝑔) → ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵))
4		addcomnq 10874	. . . . . . 7 ⊢ (𝑔 +Q ℎ) = (ℎ +Q 𝑔)
5	4	breq2i 5108	. . . . . 6 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ 𝑥 <Q (ℎ +Q 𝑔))
6	4	fveq2i 6845	. . . . . . . . 9 ⊢ (*Q‘(𝑔 +Q ℎ)) = (*Q‘(ℎ +Q 𝑔))
7	6	oveq2i 7379	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) = (𝑥 ·Q (*Q‘(ℎ +Q 𝑔)))
8	7	oveq1i 7378	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) = ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ)
9	8	eleq1i 2828	. . . . . 6 ⊢ (((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵 ↔ ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵)
10	3, 5, 9	3imtr4g 296	. . . . 5 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (𝑔 +Q ℎ) → ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) ∈ 𝐵))
11	10	adantll 715	. . . 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 512	. . 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 482	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)))
14		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝐴 ∈ P)
15		simpl 482	. . . . 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 10906	. . . . 5 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)})
18		addclnq 10868	. . . . 5 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q)
19	17, 18	genpprecl 10924	. . . 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 10884	. . . . 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 10882	. . . . 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 2762	. . . 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 10876	. . . . . . 7 ⊢ ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = ((𝑔 +Q ℎ) ·Q (*Q‘(𝑔 +Q ℎ)))
26		elprnq 10914	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
27		elprnq 10914	. . . . . . . . 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 10868	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
30		recidnq 10888	. . . . . . . 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	eqtrid 2784	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = 1Q)
33	32	oveq2d 7384	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ))) = (𝑥 ·Q 1Q))
34		mulidnq 10886	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
35	33, 34	sylan9eq 2792	. . . 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	eqtrid 2784	. . 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 2822	. 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 239	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 395   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Qcnq 10775  1_Qc1q 10776   +_Q cplq 10778   ·_Q cmq 10779  *_Qcrq 10780   <_Q cltq 10781  Pcnp 10782   +_P cpp 10784

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-omul 8412  df-er 8645  df-ni 10795  df-pli 10796  df-mi 10797  df-lti 10798  df-plpq 10831  df-mpq 10832  df-ltpq 10833  df-enq 10834  df-nq 10835  df-erq 10836  df-plq 10837  df-mq 10838  df-1nq 10839  df-rq 10840  df-ltnq 10841  df-np 10904  df-plp 10906

This theorem is referenced by:  addclpr  10941