ltaddpr - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ltaddpr		GIF version

Theorem ltaddpr 7657

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 7535	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		prml 7537	. . . 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 7535	. . . . 5 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		prarloc 7563	. . . . 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 7542	. . . . . . . . . . 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 7528	. . . . . . . . . . . . 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 7435	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
20	18, 19	genpprecll 7574	. . . . . . . . . . . 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 7597	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)
24		prop 7535	. . . . . . . . . . . 12 ⊢ ((𝐴 +_P 𝐵) ∈ P → ⟨(1^st ‘(𝐴 +_P 𝐵)), (2^nd ‘(𝐴 +_P 𝐵))⟩ ∈ P)
25		prcdnql 7544	. . . . . . . . . . . 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 2543	. . . . . . 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 1247	. . . . . 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 7566	. . . . . . . 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 2619	. . 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 2616	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 2164 ∃wrex 2473 ⟨cop 3621 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 1^st c1st 6191 2^nd c2nd 6192 Qcnq 7340 +_Q cplq 7342 <_Q cltq 7345 Pcnp 7351 +_P cpp 7353 <_P cltp 7355

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-eprel 4320 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-1o 6469 df-2o 6470 df-oadd 6473 df-omul 6474 df-er 6587 df-ec 6589 df-qs 6593 df-ni 7364 df-pli 7365 df-mi 7366 df-lti 7367 df-plpq 7404 df-mpq 7405 df-enq 7407 df-nqqs 7408 df-plqqs 7409 df-mqqs 7410 df-1nqqs 7411 df-rq 7412 df-ltnqqs 7413 df-enq0 7484 df-nq0 7485 df-0nq0 7486 df-plq0 7487 df-mq0 7488 df-inp 7526 df-iplp 7528 df-iltp 7530

This theorem is referenced by: ltexprlemrl 7670 ltaprlem 7678 ltaprg 7679 prplnqu 7680 ltmprr 7702 caucvgprprlemnkltj 7749 caucvgprprlemnkeqj 7750 caucvgprprlemnbj 7753 0lt1sr 7825 recexgt0sr 7833 mulgt0sr 7838 archsr 7842 prsrpos 7845 mappsrprg 7864 pitoregt0 7909

W3C validator