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Theorem ltaddpr 7077
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 6955 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prml 6957 . . . 4 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
31, 2syl 14 . . 3 (𝐵P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
43adantl 271 . 2 ((𝐴P𝐵P) → ∃𝑝Q 𝑝 ∈ (1st𝐵))
5 prop 6955 . . . . 5 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 prarloc 6983 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
75, 6sylan 277 . . . 4 ((𝐴P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
87ad2ant2r 493 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
9 elprnqu 6962 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
105, 9sylan 277 . . . . . . . . . 10 ((𝐴P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1110adantlr 461 . . . . . . . . 9 (((𝐴P𝐵P) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1211ad2ant2rl 495 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑞Q)
1312adantr 270 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞Q)
14 simplrr 503 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (2nd𝐴))
15 simprl 498 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑟 ∈ (1st𝐴))
16 simplr 497 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑝 ∈ (1st𝐵))
1715, 16jca 300 . . . . . . . . . . . 12 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑟 ∈ (1st𝐴) ∧ 𝑝 ∈ (1st𝐵)))
18 df-iplp 6948 . . . . . . . . . . . . 13 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
19 addclnq 6855 . . . . . . . . . . . . 13 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
2018, 19genpprecll 6994 . . . . . . . . . . . 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 434 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → ((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
23 addclpr 7017 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
24 prop 6955 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
25 prcdnql 6964 . . . . . . . . . . . 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 277 . . . . . . . . . . 11 (((𝐴 +P 𝐵) ∈ P ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2723, 26sylan 277 . . . . . . . . . 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 392 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3029imp 122 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
31 rspe 2420 . . . . . . 7 ((𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3213, 14, 30, 31syl12anc 1170 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
33 ltdfpr 6986 . . . . . . . 8 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3423, 33syldan 276 . . . . . . 7 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3534ad3antrrr 476 . . . . . 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 165 . . . . 5 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝐴<P (𝐴 +P 𝐵))
3736ex 113 . . . 4 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝐴<P (𝐴 +P 𝐵)))
3837rexlimdvva 2492 . . 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 2489 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1436  wrex 2356  cop 3428   class class class wbr 3814  cfv 4972  (class class class)co 5594  1st c1st 5847  2nd c2nd 5848  Qcnq 6760   +Q cplq 6762   <Q cltq 6765  Pcnp 6771   +P cpp 6773  <P cltp 6775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-eprel 4083  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-1o 6116  df-2o 6117  df-oadd 6120  df-omul 6121  df-er 6225  df-ec 6227  df-qs 6231  df-ni 6784  df-pli 6785  df-mi 6786  df-lti 6787  df-plpq 6824  df-mpq 6825  df-enq 6827  df-nqqs 6828  df-plqqs 6829  df-mqqs 6830  df-1nqqs 6831  df-rq 6832  df-ltnqqs 6833  df-enq0 6904  df-nq0 6905  df-0nq0 6906  df-plq0 6907  df-mq0 6908  df-inp 6946  df-iplp 6948  df-iltp 6950
This theorem is referenced by:  ltexprlemrl  7090  ltaprlem  7098  ltaprg  7099  prplnqu  7100  ltmprr  7122  caucvgprprlemnkltj  7169  caucvgprprlemnkeqj  7170  caucvgprprlemnbj  7173  0lt1sr  7232  recexgt0sr  7240  mulgt0sr  7244  archsr  7248  prsrpos  7251  pitoregt0  7307
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