ltaddpr - Intuitionistic Logic Explorer

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Theorem ltaddpr 7559

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 7437	. . . 4 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
2		prml 7439	. . . 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 7437	. . . . 5 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
6		prarloc 7465	. . . . 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 506	. . 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 7444	. . . . . . . . . . 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 474	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑞 ∈ (2^nd ‘𝐴)) → 𝑞 ∈ Q)
12	11	ad2ant2rl 508	. . . . . . . 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 531	. . . . . . 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 526	. . . . . . . . . . . . 13 ⊢ (((𝑝 ∈ Q ∧ 𝑝 ∈ (1^st ‘𝐵)) ∧ (𝑟 ∈ (1^st ‘𝐴) ∧ 𝑞 ∈ (2^nd ‘𝐴))) → 𝑟 ∈ (1^st ‘𝐴))
16		simplr 525	. . . . . . . . . . . . 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 7430	. . . . . . . . . . . . 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 7337	. . . . . . . . . . . . 13 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔 +_Q ℎ) ∈ Q)
20	18, 19	genpprecll 7476	. . . . . . . . . . . 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 443	. . . . . . . . . 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 7499	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 +_P 𝐵) ∈ P)
24		prop 7437	. . . . . . . . . . . 12 ⊢ ((𝐴 +_P 𝐵) ∈ P → ⟨(1^st ‘(𝐴 +_P 𝐵)), (2^nd ‘(𝐴 +_P 𝐵))⟩ ∈ P)
25		prcdnql 7446	. . . . . . . . . . . 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 2519	. . . . . . 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 1231	. . . . . 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 7468	. . . . . . . 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 489	. . . . . 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 2595	. . 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 2592	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 2141 ∃wrex 2449 ⟨cop 3586 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 1^st c1st 6117 2^nd c2nd 6118 Qcnq 7242 +_Q cplq 7244 <_Q cltq 7247 Pcnp 7253 +_P cpp 7255 <_P cltp 7257

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-1o 6395 df-2o 6396 df-oadd 6399 df-omul 6400 df-er 6513 df-ec 6515 df-qs 6519 df-ni 7266 df-pli 7267 df-mi 7268 df-lti 7269 df-plpq 7306 df-mpq 7307 df-enq 7309 df-nqqs 7310 df-plqqs 7311 df-mqqs 7312 df-1nqqs 7313 df-rq 7314 df-ltnqqs 7315 df-enq0 7386 df-nq0 7387 df-0nq0 7388 df-plq0 7389 df-mq0 7390 df-inp 7428 df-iplp 7430 df-iltp 7432

This theorem is referenced by: ltexprlemrl 7572 ltaprlem 7580 ltaprg 7581 prplnqu 7582 ltmprr 7604 caucvgprprlemnkltj 7651 caucvgprprlemnkeqj 7652 caucvgprprlemnbj 7655 0lt1sr 7727 recexgt0sr 7735 mulgt0sr 7740 archsr 7744 prsrpos 7747 mappsrprg 7766 pitoregt0 7811

W3C validator