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

 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 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7290 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prml 7292 . . . 4 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
31, 2syl 14 . . 3 (𝐵P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
43adantl 275 . 2 ((𝐴P𝐵P) → ∃𝑝Q 𝑝 ∈ (1st𝐵))
5 prop 7290 . . . . 5 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 prarloc 7318 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
75, 6sylan 281 . . . 4 ((𝐴P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
87ad2ant2r 500 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
9 elprnqu 7297 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
105, 9sylan 281 . . . . . . . . . 10 ((𝐴P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1110adantlr 468 . . . . . . . . 9 (((𝐴P𝐵P) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1211ad2ant2rl 502 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑞Q)
1312adantr 274 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞Q)
14 simplrr 525 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (2nd𝐴))
15 simprl 520 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑟 ∈ (1st𝐴))
16 simplr 519 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑝 ∈ (1st𝐵))
1715, 16jca 304 . . . . . . . . . . . 12 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑟 ∈ (1st𝐴) ∧ 𝑝 ∈ (1st𝐵)))
18 df-iplp 7283 . . . . . . . . . . . . 13 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
19 addclnq 7190 . . . . . . . . . . . . 13 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
2018, 19genpprecll 7329 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑟 ∈ (1st𝐴) ∧ 𝑝 ∈ (1st𝐵)) → (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
2117, 20syl5 32 . . . . . . . . . . 11 ((𝐴P𝐵P) → (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
2221imdistani 441 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → ((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
23 addclpr 7352 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
24 prop 7290 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
25 prcdnql 7299 . . . . . . . . . . . 12 ((⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2624, 25sylan 281 . . . . . . . . . . 11 (((𝐴 +P 𝐵) ∈ P ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2723, 26sylan 281 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2822, 27syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2928anassrs 397 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3029imp 123 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
31 rspe 2481 . . . . . . 7 ((𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3213, 14, 30, 31syl12anc 1214 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
33 ltdfpr 7321 . . . . . . . 8 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3423, 33syldan 280 . . . . . . 7 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3534ad3antrrr 483 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3632, 35mpbird 166 . . . . 5 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝐴<P (𝐴 +P 𝐵))
3736ex 114 . . . 4 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝐴<P (𝐴 +P 𝐵)))
3837rexlimdvva 2557 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → (∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝) → 𝐴<P (𝐴 +P 𝐵)))
398, 38mpd 13 . 2 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → 𝐴<P (𝐴 +P 𝐵))
404, 39rexlimddv 2554 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∈ wcel 1480  ∃wrex 2417  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097
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