ltaddpr - Intuitionistic Logic Explorer

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Theorem ltaddpr 7519

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 7397	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		prml 7399	. . . 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 275	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑝 ∈ Q 𝑝 ∈ (1^st ‘𝐵))
5		prop 7397	. . . . 5 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		prarloc 7425	. . . . 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 281	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝑝 ∈ Q) → ∃𝑟 ∈ (1^st ‘𝐴)∃𝑞 ∈ (2^nd ‘𝐴)𝑞 <_Q (𝑟 +_Q 𝑝))
8	7	ad2ant2r 501	. . 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 7404	. . . . . . . . . . 11 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
10	5, 9	sylan 281	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
11	10	adantlr 469	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
12	11	ad2ant2rl 503	. . . . . . . 8 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵))) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑞 ∈ Q)
13	12	adantr 274	. . . . . . 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 526	. . . . . . 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 521	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑟 ∈ (1^st ‘𝐴))
16		simplr 520	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑝 ∈ (1^st ‘𝐵))
17	15, 16	jca 304	. . . . . . . . . . . 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 7390	. . . . . . . . . . . . 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 7297	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
20	18, 19	genpprecll 7436	. . . . . . . . . . . 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 442	. . . . . . . . . 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 7459	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)
24		prop 7397	. . . . . . . . . . . 12 ⊢ ((𝐴 +_P 𝐵) ∈ P → ⟨(1^st ‘(𝐴 +_P 𝐵)), (2^nd ‘(𝐴 +_P 𝐵))⟩ ∈ P)
25		prcdnql 7406	. . . . . . . . . . . 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 281	. . . . . . . . . . 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 281	. . . . . . . . . 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 398	. . . . . . . 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 123	. . . . . . 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 2506	. . . . . . 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 1218	. . . . . 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 7428	. . . . . . . 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 280	. . . . . . 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 484	. . . . . 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 166	. . . . 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 114	. . . 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 2582	. . 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 2579	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → 𝐴<_P (𝐴 +_P 𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∈ wcel 2128 ∃wrex 2436 ⟨cop 3564 class class class wbr 3967 ‘cfv 5172 (class class class)co 5826 1^st c1st 6088 2^nd c2nd 6089 Qcnq 7202 +_Q cplq 7204 <_Q cltq 7207 Pcnp 7213 +_P cpp 7215 <_P cltp 7217

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4081 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-br 3968 df-opab 4028 df-mpt 4029 df-tr 4065 df-eprel 4251 df-id 4255 df-po 4258 df-iso 4259 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 df-1st 6090 df-2nd 6091 df-recs 6254 df-irdg 6319 df-1o 6365 df-2o 6366 df-oadd 6369 df-omul 6370 df-er 6482 df-ec 6484 df-qs 6488 df-ni 7226 df-pli 7227 df-mi 7228 df-lti 7229 df-plpq 7266 df-mpq 7267 df-enq 7269 df-nqqs 7270 df-plqqs 7271 df-mqqs 7272 df-1nqqs 7273 df-rq 7274 df-ltnqqs 7275 df-enq0 7346 df-nq0 7347 df-0nq0 7348 df-plq0 7349 df-mq0 7350 df-inp 7388 df-iplp 7390 df-iltp 7392

This theorem is referenced by: ltexprlemrl 7532 ltaprlem 7540 ltaprg 7541 prplnqu 7542 ltmprr 7564 caucvgprprlemnkltj 7611 caucvgprprlemnkeqj 7612 caucvgprprlemnbj 7615 0lt1sr 7687 recexgt0sr 7695 mulgt0sr 7700 archsr 7704 prsrpos 7707 mappsrprg 7726 pitoregt0 7771

W3C validator