Theorem ltaddpr 9735

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 9690	. . . . 5 ⊢ (𝐵 ∈ P → 𝐵 ≠ ∅)
2		n0 3890	. . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
3	1, 2	sylib 207	. . . 4 ⊢ (𝐵 ∈ P → ∃𝑦 𝑦 ∈ 𝐵)
4	3	adantl 481	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑦 𝑦 ∈ 𝐵)
5		addclpr 9719	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +P 𝐵) ∈ P)
6	5	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐴 +P 𝐵) ∈ P)
7		df-plp 9684	. . . . . . . . . . . . 13 ⊢ +P = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦 +Q 𝑧)})
8		addclnq 9646	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 +Q 𝑧) ∈ Q)
9	7, 8	genpprecl 9702	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
10	9	imp 444	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
11		elprnq 9692	. . . . . . . . . . . . 13 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
12		addnqf 9649	. . . . . . . . . . . . . . 15 ⊢ +Q :(Q × Q)⟶Q
13	12	fdmi 5965	. . . . . . . . . . . . . 14 ⊢ dom +Q = (Q × Q)
14		0nnq 9625	. . . . . . . . . . . . . 14 ⊢ ¬ ∅ ∈ Q
15	13, 14	ndmovrcl 6718	. . . . . . . . . . . . 13 ⊢ ((𝑥 +Q 𝑦) ∈ Q → (𝑥 ∈ Q ∧ 𝑦 ∈ Q))
16		ltaddnq 9675	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦))
17	11, 15, 16	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 <Q (𝑥 +Q 𝑦))
18		prcdnq 9694	. . . . . . . . . . . 12 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 <Q (𝑥 +Q 𝑦) → 𝑥 ∈ (𝐴 +P 𝐵)))
19	17, 18	mpd 15	. . . . . . . . . . 11 ⊢ (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
20	6, 10, 19	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
21	20	exp32 629	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → 𝑥 ∈ (𝐴 +P 𝐵))))
22	21	com23 84	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))))
23	22	alrimdv 1844	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵))))
24		dfss2 3557	. . . . . . 7 ⊢ (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ (𝐴 +P 𝐵)))
25	23, 24	syl6ibr 241	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → 𝐴 ⊆ (𝐴 +P 𝐵)))
26		vex 3176	. . . . . . . . 9 ⊢ 𝑦 ∈ V
27	26	prlem934 9734	. . . . . . . 8 ⊢ (𝐴 ∈ P → ∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
28	27	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
29		eleq2 2677	. . . . . . . . . . . . 13 ⊢ (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
30	29	biimprcd 239	. . . . . . . . . . . 12 ⊢ ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴))
31	30	con3d 147	. . . . . . . . . . 11 ⊢ ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))
32	9, 31	syl6 34	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))
33	32	expd 451	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
34	33	com34 89	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ 𝐴 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
35	34	rexlimdv 3012	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑥 ∈ 𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
36	28, 35	mpd 15	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
37	25, 36	jcad 554	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
38		dfpss2 3654	. . . . 5 ⊢ (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
39	37, 38	syl6ibr 241	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵)))
40	39	exlimdv 1848	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑦 𝑦 ∈ 𝐵 → 𝐴 ⊊ (𝐴 +P 𝐵)))
41	4, 40	mpd 15	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴 ⊊ (𝐴 +P 𝐵))
42		ltprord 9731	. . 3 ⊢ ((𝐴 ∈ P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
43	5, 42	syldan 486	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
44	41, 43	mpbird 246	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<P (𝐴 +P 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874   class class class wbr 4583   × cxp 5036  (class class class)co 6549  Qcnq 9553   +_Q cplq 9556   <_Q cltq 9559  Pcnp 9560   +_P cpp 9562  <_P cltp 9564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686

This theorem is referenced by:  ltaddpr2  9736  ltexprlem7  9743  ltaprlem  9745  0lt1sr  9795  mappsrpr  9808