Theorem ltaddpr 3934

Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123.

Assertion

Ref	Expression
ltaddpr	⊢ ((A ∈ P ∧ B ∈ P) → A<P (A +P B))

Proof of Theorem ltaddpr

Step	Hyp	Ref	Expression
1		prn0 3887	. . . . 5 ⊢ (B ∈ P → ¬ B = ∅)
2		n0 1714	. . . . 5 ⊢ (¬ B = ∅ ↔ ∃y y ∈ B)
3	1, 2	sylib 173	. . . 4 ⊢ (B ∈ P → ∃y y ∈ B)
4	3	adantl 305	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → ∃y y ∈ B)
5		elprpq 3889	. . . . . . . . . . . . 13 ⊢ (((A +P B) ∈ P ∧ (x +Q y) ∈ (A +P B)) → (x +Q y) ∈ Q)
6		visset 1350	. . . . . . . . . . . . . 14 ⊢ y ∈ V
7		dmaddpq 3853	. . . . . . . . . . . . . 14 ⊢ dom +Q = (Q × Q)
8		0npq 3844	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ ∈ Q
9	6, 7, 8	ndmoprrcl 3060	. . . . . . . . . . . . 13 ⊢ ((x +Q y) ∈ Q → (x ∈ Q ∧ y ∈ Q))
10		visset 1350	. . . . . . . . . . . . . 14 ⊢ x ∈ V
11	10, 6	ltaddpq 3873	. . . . . . . . . . . . 13 ⊢ ((x ∈ Q ∧ y ∈ Q) → x <Q (x +Q y))
12	5, 9, 11	3syl 21	. . . . . . . . . . . 12 ⊢ (((A +P B) ∈ P ∧ (x +Q y) ∈ (A +P B)) → x <Q (x +Q y))
13		prcdpq 3891	. . . . . . . . . . . 12 ⊢ (((A +P B) ∈ P ∧ (x +Q y) ∈ (A +P B)) → (x <Q (x +Q y) → x ∈ (A +P B)))
14	12, 13	mpd 46	. . . . . . . . . . 11 ⊢ (((A +P B) ∈ P ∧ (x +Q y) ∈ (A +P B)) → x ∈ (A +P B))
15		addclpr 3914	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ B ∈ P) → (A +P B) ∈ P)
16	15	adantr 306	. . . . . . . . . . 11 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (x ∈ A ∧ y ∈ B)) → (A +P B) ∈ P)
17		df-plp 3882	. . . . . . . . . . . . 13 ⊢ +P = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃y ∈ w ∃z ∈ v x = (y +Q z)})}
18	17	genpprecl 3898	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ B ∈ P) → ((x ∈ A ∧ y ∈ B) → (x +Q y) ∈ (A +P B)))
19	18	imp 277	. . . . . . . . . . 11 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (x ∈ A ∧ y ∈ B)) → (x +Q y) ∈ (A +P B))
20	14, 16, 19	sylanc 361	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (x ∈ A ∧ y ∈ B)) → x ∈ (A +P B))
21	20	exp32 294	. . . . . . . . 9 ⊢ ((A ∈ P ∧ B ∈ P) → (x ∈ A → (y ∈ B → x ∈ (A +P B))))
22	21	com23 32	. . . . . . . 8 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → (x ∈ A → x ∈ (A +P B))))
23	22	19.21adv 945	. . . . . . 7 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → ∀x(x ∈ A → x ∈ (A +P B))))
24		dfss2 1497	. . . . . . 7 ⊢ (A ⊆ (A +P B) ↔ ∀x(x ∈ A → x ∈ (A +P B)))
25	23, 24	syl6ibr 186	. . . . . 6 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → A ⊆ (A +P B)))
26		eleq2 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ (A = (A +P B) → ((x +Q y) ∈ A ↔ (x +Q y) ∈ (A +P B)))
27	26	biimprcd 138	. . . . . . . . . . . . . . . . . 18 ⊢ ((x +Q y) ∈ (A +P B) → (A = (A +P B) → (x +Q y) ∈ A))
28	27	con3d 87	. . . . . . . . . . . . . . . . 17 ⊢ ((x +Q y) ∈ (A +P B) → (¬ (x +Q y) ∈ A → ¬ A = (A +P B)))
29	18, 28	syl6 23	. . . . . . . . . . . . . . . 16 ⊢ ((A ∈ P ∧ B ∈ P) → ((x ∈ A ∧ y ∈ B) → (¬ (x +Q y) ∈ A → ¬ A = (A +P B))))
30	29	exp3a 292	. . . . . . . . . . . . . . 15 ⊢ ((A ∈ P ∧ B ∈ P) → (x ∈ A → (y ∈ B → (¬ (x +Q y) ∈ A → ¬ A = (A +P B)))))
31	30	com34 36	. . . . . . . . . . . . . 14 ⊢ ((A ∈ P ∧ B ∈ P) → (x ∈ A → (¬ (x +Q y) ∈ A → (y ∈ B → ¬ A = (A +P B)))))
32	31	imp3a 279	. . . . . . . . . . . . 13 ⊢ ((A ∈ P ∧ B ∈ P) → ((x ∈ A ∧ ¬ (x +Q y) ∈ A) → (y ∈ B → ¬ A = (A +P B))))
33	32	19.23adv 954	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ B ∈ P) → (∃x(x ∈ A ∧ ¬ (x +Q y) ∈ A) → (y ∈ B → ¬ A = (A +P B))))
34		prlem934 3933	. . . . . . . . . . . 12 ⊢ ((A ∈ P ∧ y ∈ Q) → ∃x(x ∈ A ∧ ¬ (x +Q y) ∈ A))
35	33, 34	syl5 22	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ B ∈ P) → ((A ∈ P ∧ y ∈ Q) → (y ∈ B → ¬ A = (A +P B))))
36		elprpq 3889	. . . . . . . . . . 11 ⊢ ((B ∈ P ∧ y ∈ B) → y ∈ Q)
37	35, 36	sylan2i 357	. . . . . . . . . 10 ⊢ ((A ∈ P ∧ B ∈ P) → ((A ∈ P ∧ (B ∈ P ∧ y ∈ B)) → (y ∈ B → ¬ A = (A +P B))))
38	37	exp4d 298	. . . . . . . . 9 ⊢ ((A ∈ P ∧ B ∈ P) → (A ∈ P → (B ∈ P → (y ∈ B → (y ∈ B → ¬ A = (A +P B))))))
39	38	imp3a 279	. . . . . . . 8 ⊢ ((A ∈ P ∧ B ∈ P) → ((A ∈ P ∧ B ∈ P) → (y ∈ B → (y ∈ B → ¬ A = (A +P B)))))
40	39	pm2.43i 58	. . . . . . 7 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → (y ∈ B → ¬ A = (A +P B))))
41	40	pm2.43d 59	. . . . . 6 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → ¬ A = (A +P B)))
42	25, 41	jcad 455	. . . . 5 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → (A ⊆ (A +P B) ∧ ¬ A = (A +P B))))
43		dfpss2 1557	. . . . 5 ⊢ (A ⊂ (A +P B) ↔ (A ⊆ (A +P B) ∧ ¬ A = (A +P B)))
44	42, 43	syl6ibr 186	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → (y ∈ B → A ⊂ (A +P B)))
45	44	19.23adv 954	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → (∃y y ∈ B → A ⊂ (A +P B)))
46	4, 45	mpd 46	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → A ⊂ (A +P B))
47		ltprord 3928	. . . 4 ⊢ ((A ∈ P ∧ (A +P B) ∈ P) → (A<P (A +P B) ↔ A ⊂ (A +P B)))
48	47, 15	sylan2 346	. . 3 ⊢ ((A ∈ P ∧ (A ∈ P ∧ B ∈ P)) → (A<P (A +P B) ↔ A ⊂ (A +P B)))
49	48	anabss5 384	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (A<P (A +P B) ↔ A ⊂ (A +P B)))
50	46, 49	mpbird 171	1 ⊢ ((A ∈ P ∧ B ∈ P) → A<P (A +P B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   ⊂ wpss 1488  ∅c0 1707   class class class wbr 2054  (class class class)co 3001  Qcnq 3773   +_Q cplq 3775   <_Q cltq 3778  Pcnp 3779   +_P cpp 3781  <_P cltp 3783

This theorem is referenced by:  ltaddpr2 3935  ltexprlem7 3942  ltaprlem 3944  0lt1sr 3998  mappsrpr 4012

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-ltp 3884

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