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Theorem ltprordil 6422
Description: If a positive real is less than a second positive real, its lower cut is a subset of the second's lower cut. (Contributed by Jim Kingdon, 23-Dec-2019.)
Assertion
Ref Expression
ltprordil (A<P B → (1stA) ⊆ (1stB))

Proof of Theorem ltprordil
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 6353 . . . 4 <P ⊆ (P × P)
21brel 4315 . . 3 (A<P B → (A P B P))
3 ltdfpr 6354 . . . 4 ((A P B P) → (A<P Bx Q (x (2ndA) x (1stB))))
43biimpd 132 . . 3 ((A P B P) → (A<P Bx Q (x (2ndA) x (1stB))))
52, 4mpcom 32 . 2 (A<P Bx Q (x (2ndA) x (1stB)))
6 simpll 469 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → A<P B)
7 simpr 103 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → y (1stA))
8 simprrl 479 . . . . . . 7 ((A<P B (x Q (x (2ndA) x (1stB)))) → x (2ndA))
98adantr 261 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → x (2ndA))
102simpld 105 . . . . . . . 8 (A<P BA P)
11 prop 6323 . . . . . . . 8 (A P → ⟨(1stA), (2ndA)⟩ P)
1210, 11syl 14 . . . . . . 7 (A<P B → ⟨(1stA), (2ndA)⟩ P)
13 prltlu 6335 . . . . . . 7 ((⟨(1stA), (2ndA)⟩ P y (1stA) x (2ndA)) → y <Q x)
1412, 13syl3an1 1152 . . . . . 6 ((A<P B y (1stA) x (2ndA)) → y <Q x)
156, 7, 9, 14syl3anc 1119 . . . . 5 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → y <Q x)
16 simprrr 480 . . . . . . 7 ((A<P B (x Q (x (2ndA) x (1stB)))) → x (1stB))
1716adantr 261 . . . . . 6 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → x (1stB))
182simprd 107 . . . . . . . 8 (A<P BB P)
19 prop 6323 . . . . . . . 8 (B P → ⟨(1stB), (2ndB)⟩ P)
2018, 19syl 14 . . . . . . 7 (A<P B → ⟨(1stB), (2ndB)⟩ P)
21 prcdnql 6332 . . . . . . 7 ((⟨(1stB), (2ndB)⟩ P x (1stB)) → (y <Q xy (1stB)))
2220, 21sylan 267 . . . . . 6 ((A<P B x (1stB)) → (y <Q xy (1stB)))
236, 17, 22syl2anc 393 . . . . 5 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → (y <Q xy (1stB)))
2415, 23mpd 13 . . . 4 (((A<P B (x Q (x (2ndA) x (1stB)))) y (1stA)) → y (1stB))
2524ex 108 . . 3 ((A<P B (x Q (x (2ndA) x (1stB)))) → (y (1stA) → y (1stB)))
2625ssrdv 2924 . 2 ((A<P B (x Q (x (2ndA) x (1stB)))) → (1stA) ⊆ (1stB))
275, 26rexlimddv 2411 1 (A<P B → (1stA) ⊆ (1stB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1370  wrex 2281  wss 2890  cop 3349   class class class wbr 3734  cfv 4825  1st c1st 5684  2nd c2nd 5685  Qcnq 6134   <Q cltq 6139  Pcnp 6145  <P cltp 6149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234
This theorem depends on definitions:  df-bi 110  df-dc 731  df-3or 872  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-tr 3825  df-eprel 3996  df-id 4000  df-po 4003  df-iso 4004  df-iord 4048  df-on 4050  df-suc 4053  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-recs 5838  df-irdg 5874  df-oadd 5916  df-omul 5917  df-er 6013  df-ec 6015  df-qs 6019  df-ni 6158  df-mi 6160  df-lti 6161  df-enq 6200  df-nqqs 6201  df-ltnqqs 6206  df-inp 6314  df-iltp 6318
This theorem is referenced by:  ltexprlemrl  6441
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