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Theorem ltaddpr 7595
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))

Proof of Theorem ltaddpr
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7473 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prml 7475 . . . 4 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
31, 2syl 14 . . 3 (𝐵P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
43adantl 277 . 2 ((𝐴P𝐵P) → ∃𝑝Q 𝑝 ∈ (1st𝐵))
5 prop 7473 . . . . 5 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 prarloc 7501 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
75, 6sylan 283 . . . 4 ((𝐴P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
87ad2ant2r 509 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
9 elprnqu 7480 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
105, 9sylan 283 . . . . . . . . . 10 ((𝐴P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1110adantlr 477 . . . . . . . . 9 (((𝐴P𝐵P) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1211ad2ant2rl 511 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑞Q)
1312adantr 276 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞Q)
14 simplrr 536 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (2nd𝐴))
15 simprl 529 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑟 ∈ (1st𝐴))
16 simplr 528 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑝 ∈ (1st𝐵))
1715, 16jca 306 . . . . . . . . . . . 12 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑟 ∈ (1st𝐴) ∧ 𝑝 ∈ (1st𝐵)))
18 df-iplp 7466 . . . . . . . . . . . . 13 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
19 addclnq 7373 . . . . . . . . . . . . 13 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
2018, 19genpprecll 7512 . . . . . . . . . . . 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 445 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → ((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
23 addclpr 7535 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
24 prop 7473 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
25 prcdnql 7482 . . . . . . . . . . . 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 283 . . . . . . . . . . 11 (((𝐴 +P 𝐵) ∈ P ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2723, 26sylan 283 . . . . . . . . . 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 400 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3029imp 124 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
31 rspe 2526 . . . . . . 7 ((𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3213, 14, 30, 31syl12anc 1236 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
33 ltdfpr 7504 . . . . . . . 8 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3423, 33syldan 282 . . . . . . 7 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3534ad3antrrr 492 . . . . . 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 167 . . . . 5 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝐴<P (𝐴 +P 𝐵))
3736ex 115 . . . 4 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝐴<P (𝐴 +P 𝐵)))
3837rexlimdvva 2602 . . 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 2599 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2148  wrex 2456  cop 3595   class class class wbr 4003  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  Qcnq 7278   +Q cplq 7280   <Q cltq 7283  Pcnp 7289   +P cpp 7291  <P cltp 7293
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-iplp 7466  df-iltp 7468
This theorem is referenced by:  ltexprlemrl  7608  ltaprlem  7616  ltaprg  7617  prplnqu  7618  ltmprr  7640  caucvgprprlemnkltj  7687  caucvgprprlemnkeqj  7688  caucvgprprlemnbj  7691  0lt1sr  7763  recexgt0sr  7771  mulgt0sr  7776  archsr  7780  prsrpos  7783  mappsrprg  7802  pitoregt0  7847
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