Theorem addclprlem2 11012

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 11011	. . . . 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 11011	. . . . . 6 ⊢ (((𝐵 ∈ P ∧ ℎ ∈ 𝐵) ∧ 𝑥 ∈ Q) → (𝑥 <Q (ℎ +Q 𝑔) → ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ) ∈ 𝐵))
4		addcomnq 10946	. . . . . . 7 ⊢ (𝑔 +Q ℎ) = (ℎ +Q 𝑔)
5	4	breq2i 5157	. . . . . 6 ⊢ (𝑥 <Q (𝑔 +Q ℎ) ↔ 𝑥 <Q (ℎ +Q 𝑔))
6	4	fveq2i 6895	. . . . . . . . 9 ⊢ (*Q‘(𝑔 +Q ℎ)) = (*Q‘(ℎ +Q 𝑔))
7	6	oveq2i 7420	. . . . . . . 8 ⊢ (𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) = (𝑥 ·Q (*Q‘(ℎ +Q 𝑔)))
8	7	oveq1i 7419	. . . . . . 7 ⊢ ((𝑥 ·Q (*Q‘(𝑔 +Q ℎ))) ·Q ℎ) = ((𝑥 ·Q (*Q‘(ℎ +Q 𝑔))) ·Q ℎ)
9	8	eleq1i 2825	. . . . . 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 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 514	. . 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 484	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) ∧ 𝑥 ∈ Q) → ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)))
14		simpl 484	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝐴 ∈ P)
15		simpl 484	. . . . 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 10978	. . . . 5 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)})
18		addclnq 10940	. . . . 5 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q)
19	17, 18	genpprecl 10996	. . . 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 10956	. . . . 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 10954	. . . . 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 2763	. . . 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 10948	. . . . . . 7 ⊢ ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = ((𝑔 +Q ℎ) ·Q (*Q‘(𝑔 +Q ℎ)))
26		elprnq 10986	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → 𝑔 ∈ Q)
27		elprnq 10986	. . . . . . . . 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 10940	. . . . . . . 8 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +Q ℎ) ∈ Q)
30		recidnq 10960	. . . . . . . 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 2785	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ)) = 1Q)
33	32	oveq2d 7425	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑥 ·Q ((*Q‘(𝑔 +Q ℎ)) ·Q (𝑔 +Q ℎ))) = (𝑥 ·Q 1Q))
34		mulidnq 10958	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑥 ·Q 1Q) = 𝑥)
35	33, 34	sylan9eq 2793	. . . 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 2785	. . 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 2819	. 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 238	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 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409  Qcnq 10847  1_Qc1q 10848   +_Q cplq 10850   ·_Q cmq 10851  *_Qcrq 10852   <_Q cltq 10853  Pcnp 10854   +_P cpp 10856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-ni 10867  df-pli 10868  df-mi 10869  df-lti 10870  df-plpq 10903  df-mpq 10904  df-ltpq 10905  df-enq 10906  df-nq 10907  df-erq 10908  df-plq 10909  df-mq 10910  df-1nq 10911  df-rq 10912  df-ltnq 10913  df-np 10976  df-plp 10978

This theorem is referenced by:  addclpr  11013