Theorem ltaddpr 10979

Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ltaddpr

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

Step	Hyp	Ref	Expression
1		prn0 10934	. . . . 5 ⊢ (𝐵 ∈ P → 𝐵 ≠ ∅)
2		n0 4311	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
3	1, 2	sylib 217	. . . 4 ⊢ (𝐵 ∈ P → ∃𝑦 𝑦 ∈ 𝐵)
4	3	adantl 482	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑦 𝑦 ∈ 𝐵)
5		addclpr 10963	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
6		df-plp 10928	. . . . . . . . . . . . 13 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)})
7		addclnq 10890	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q)
8	6, 7	genpprecl 10946	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
9	8	imp 407	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
10		elprnq 10936	. . . . . . . . . . . . 13 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
11		addnqf 10893	. . . . . . . . . . . . . . 15 ⊢ +Q :(Q × Q)⟶Q
12	11	fdmi 6685	. . . . . . . . . . . . . 14 ⊢ dom +Q = (Q × Q)
13		0nnq 10869	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ ∈ Q
14	12, 13	ndmovrcl 7545	. . . . . . . . . . . . 13 ⊢ ((𝑥 +Q 𝑦) ∈ Q → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
15		ltaddnq 10919	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦))
16	10, 14, 15	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 <Q (𝑥 +Q 𝑦))
17		prcdnq 10938	. . . . . . . . . . . 12 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 <Q (𝑥 +Q 𝑦) → 𝑥 ∈ (𝐴 +P 𝐵)))
18	16, 17	mpd 15	. . . . . . . . . . 11 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
19	5, 9, 18	syl2an2r 683	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
20	19	exp32 421	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 ∈ (𝐴 +P 𝐵))))
21	20	com23 86	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))))
22	21	alrimdv 1932	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))))
23		dfss2 3933	. . . . . . 7 ⊢ (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))
24	22, 23	syl6ibr 251	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → 𝐴 ⊆ (𝐴 +P 𝐵)))
25		vex 3450	. . . . . . . . 9 ⊢ 𝑦 ∈ V
26	25	prlem934 10978	. . . . . . . 8 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
27	26	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
28		eleq2 2821	. . . . . . . . . . . . 13 ⊢ (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
29	28	biimprcd 249	. . . . . . . . . . . 12 ⊢ ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴))
30	29	con3d 152	. . . . . . . . . . 11 ⊢ ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))
31	8, 30	syl6 35	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))
32	31	expd 416	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
33	32	com34 91	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ 𝐴 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
34	33	rexlimdv 3146	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
35	27, 34	mpd 15	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
36	24, 35	jcad 513	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
37		dfpss2 4050	. . . . 5 ⊢ (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
38	36, 37	syl6ibr 251	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵)))
39	38	exlimdv 1936	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑦 𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵)))
40	4, 39	mpd 15	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴 ⊊ (𝐴 +P 𝐵))
41		ltprord 10975	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
42	5, 41	syldan 591	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
43	40, 42	mpbird 256	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<P (𝐴 +P 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2939  ∃wrex 3069   ⊆ wss 3913   ⊊ wpss 3914  ∅c0 4287   class class class wbr 5110   × cxp 5636  (class class class)co 7362  Qcnq 10797   +_Q cplq 10800   <_Q cltq 10803  Pcnp 10804   +_P cpp 10806  <_P cltp 10808

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-ni 10817  df-pli 10818  df-mi 10819  df-lti 10820  df-plpq 10853  df-mpq 10854  df-ltpq 10855  df-enq 10856  df-nq 10857  df-erq 10858  df-plq 10859  df-mq 10860  df-1nq 10861  df-rq 10862  df-ltnq 10863  df-np 10926  df-plp 10928  df-ltp 10930

This theorem is referenced by:  ltaddpr2  10980  ltexprlem7  10987  ltaprlem  10989  0lt1sr  11040  mappsrpr  11053