ltaddpr - Intuitionistic Logic Explorer

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Theorem ltaddpr 7795

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.)

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

Proof of Theorem ltaddpr Dummy variables 𝑓 𝑔 ℎ 𝑥 𝑦 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7673	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		prml 7675	. . . 4 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘𝐵))
3	1, 2	syl 14	. . 3 ⊢ (𝐵 ∈ P → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘𝐵))
4	3	adantl 277	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘𝐵))
5		prop 7673	. . . . 5 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		prarloc 7701	. . . . 5 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑝 ∈ Q) → ∃𝑟 ∈ (1^st ‘𝐴)∃𝑞 ∈ (2^nd ‘𝐴)𝑞 <_Q (𝑟 +_Q 𝑝))
7	5, 6	sylan 283	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑝 ∈ Q) → ∃𝑟 ∈ (1^st ‘𝐴)∃𝑞 ∈ (2^nd ‘𝐴)𝑞 <_Q (𝑟 +_Q 𝑝))
8	7	ad2ant2r 509	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) → ∃𝑟 ∈ (1^st ‘𝐴)∃𝑞 ∈ (2^nd ‘𝐴)𝑞 <_Q (𝑟 +_Q 𝑝))
9		elprnqu 7680	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
10	5, 9	sylan 283	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
11	10	adantlr 477	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
12	11	ad2ant2rl 511	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑞 ∈ Q)
13	12	adantr 276	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝑞 ∈ Q)
14		simplrr 536	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝑞 ∈ (2^nd ‘𝐴))
15		simprl 529	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑟 ∈ (1^st ‘𝐴))
16		simplr 528	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑝 ∈ (1^st ‘𝐵))
17	15, 16	jca 306	. . . . . . . . . . . 12 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑝 ∈ (1^st ‘𝐵)))
18		df-iplp 7666	. . . . . . . . . . . . 13 ⊢ +_P = (𝑥 ∈ P, 𝑦 ∈ P ↦ ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (1^st ‘𝑥) ∧ ℎ ∈ (1^st ‘𝑦) ∧ 𝑓 = (𝑔 +_Q ℎ))}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ Q ∃ℎ ∈ Q (𝑔 ∈ (2^nd ‘𝑥) ∧ ℎ ∈ (2^nd ‘𝑦) ∧ 𝑓 = (𝑔 +_Q ℎ))}⟩)
19		addclnq 7573	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
20	18, 19	genpprecll 7712	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑟 ∈ (1^st ‘𝐴) ∧ 𝑝 ∈ (1^st ‘𝐵)) → (𝑟 +_Q 𝑝) ∈ (1^st ‘(𝐴 +_P 𝐵))))
21	17, 20	syl5 32	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → (𝑟 +_Q 𝑝) ∈ (1^st ‘(𝐴 +_P 𝐵))))
22	21	imdistani 445	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴)))) → ((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(𝐴 +_P 𝐵))))
23		addclpr 7735	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)
24		prop 7673	. . . . . . . . . . . 12 ⊢ ((𝐴 +_P 𝐵) ∈ P → ⟨(1^st ‘(𝐴 +_P 𝐵)), (2^nd ‘(𝐴 +_P 𝐵))⟩ ∈ P)
25		prcdnql 7682	. . . . . . . . . . . 12 ⊢ ((⟨(1^st ‘(𝐴 +_P 𝐵)), (2^nd ‘(𝐴 +_P 𝐵))⟩ ∈ P ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(𝐴 +_P 𝐵))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
26	24, 25	sylan 283	. . . . . . . . . . 11 ⊢ (((𝐴 +_P 𝐵) ∈ P ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(𝐴 +_P 𝐵))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
27	23, 26	sylan 283	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑟 +_Q 𝑝) ∈ (1^st ‘(𝐴 +_P 𝐵))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
28	22, 27	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ ((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴)))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
29	28	anassrs 400	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
30	29	imp 124	. . . . . . 7 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))
31		rspe 2579	. . . . . . 7 ⊢ ((𝑞 ∈ Q ∧ (𝑞 ∈ (2^nd ‘𝐴) ∧ 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))) → ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘𝐴) ∧ 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
32	13, 14, 30, 31	syl12anc 1269	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘𝐴) ∧ 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵))))
33		ltdfpr 7704	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ (𝐴 +_P 𝐵) ∈ P) → (𝐴<_P (𝐴 +_P 𝐵) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘𝐴) ∧ 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))))
34	23, 33	syldan 282	. . . . . . 7 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴<_P (𝐴 +_P 𝐵) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘𝐴) ∧ 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))))
35	34	ad3antrrr 492	. . . . . 6 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → (𝐴<_P (𝐴 +_P 𝐵) ↔ ∃𝑞 ∈ Q (𝑞 ∈ (2^nd ‘𝐴) ∧ 𝑞 ∈ (1^st ‘(𝐴 +_P 𝐵)))))
36	32, 35	mpbird 167	. . . . 5 ⊢ (((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) ∧ 𝑞 <_Q (𝑟 +_Q 𝑝)) → 𝐴<_P (𝐴 +_P 𝐵))
37	36	ex 115	. . . 4 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → (𝑞 <_Q (𝑟 +_Q 𝑝) → 𝐴<_P (𝐴 +_P 𝐵)))
38	37	rexlimdvva 2656	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) → (∃𝑟 ∈ (1^st ‘𝐴)∃𝑞 ∈ (2^nd ‘𝐴)𝑞 <_Q (𝑟 +_Q 𝑝) → 𝐴<_P (𝐴 +_P 𝐵)))
39	8, 38	mpd 13	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) → 𝐴<_P (𝐴 +_P 𝐵))
40	4, 39	rexlimddv 2653	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<_P (𝐴 +_P 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2200 ∃wrex 2509 ⟨cop 3669 class class class wbr 4083 ‘cfv 5318 (class class class)co 6007 1^st c1st 6290 2^nd c2nd 6291 Qcnq 7478 +_Q cplq 7480 <_Q cltq 7483 Pcnp 7489 +_P cpp 7491 <_P cltp 7493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-eprel 4380 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-irdg 6522 df-1o 6568 df-2o 6569 df-oadd 6572 df-omul 6573 df-er 6688 df-ec 6690 df-qs 6694 df-ni 7502 df-pli 7503 df-mi 7504 df-lti 7505 df-plpq 7542 df-mpq 7543 df-enq 7545 df-nqqs 7546 df-plqqs 7547 df-mqqs 7548 df-1nqqs 7549 df-rq 7550 df-ltnqqs 7551 df-enq0 7622 df-nq0 7623 df-0nq0 7624 df-plq0 7625 df-mq0 7626 df-inp 7664 df-iplp 7666 df-iltp 7668

This theorem is referenced by: ltexprlemrl 7808 ltaprlem 7816 ltaprg 7817 prplnqu 7818 ltmprr 7840 caucvgprprlemnkltj 7887 caucvgprprlemnkeqj 7888 caucvgprprlemnbj 7891 0lt1sr 7963 recexgt0sr 7971 mulgt0sr 7976 archsr 7980 prsrpos 7983 mappsrprg 8002 pitoregt0 8047

W3C validator