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Theorem ltexprlemell 6570
 Description: Element in lower cut of the constructed difference. Lemma for ltexpri 6585. (Contributed by Jim Kingdon, 21-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
Assertion
Ref Expression
ltexprlemell (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
Distinct variable groups:   x,y,𝑞,A   x,B,y,𝑞   x,𝐶,y,𝑞

Proof of Theorem ltexprlemell
StepHypRef Expression
1 oveq2 5463 . . . . 5 (x = 𝑞 → (y +Q x) = (y +Q 𝑞))
21eleq1d 2103 . . . 4 (x = 𝑞 → ((y +Q x) (1stB) ↔ (y +Q 𝑞) (1stB)))
32anbi2d 437 . . 3 (x = 𝑞 → ((y (2ndA) (y +Q x) (1stB)) ↔ (y (2ndA) (y +Q 𝑞) (1stB))))
43exbidv 1703 . 2 (x = 𝑞 → (y(y (2ndA) (y +Q x) (1stB)) ↔ y(y (2ndA) (y +Q 𝑞) (1stB))))
5 ltexprlem.1 . . . 4 𝐶 = ⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩
65fveq2i 5124 . . 3 (1st𝐶) = (1st ‘⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩)
7 nqex 6347 . . . . 5 Q V
87rabex 3892 . . . 4 {x Qy(y (2ndA) (y +Q x) (1stB))} V
97rabex 3892 . . . 4 {x Qy(y (1stA) (y +Q x) (2ndB))} V
108, 9op1st 5715 . . 3 (1st ‘⟨{x Qy(y (2ndA) (y +Q x) (1stB))}, {x Qy(y (1stA) (y +Q x) (2ndB))}⟩) = {x Qy(y (2ndA) (y +Q x) (1stB))}
116, 10eqtri 2057 . 2 (1st𝐶) = {x Qy(y (2ndA) (y +Q x) (1stB))}
124, 11elrab2 2694 1 (𝑞 (1st𝐶) ↔ (𝑞 Q y(y (2ndA) (y +Q 𝑞) (1stB))))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {crab 2304  ⟨cop 3370  ‘cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266 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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-1st 5709  df-qs 6048  df-ni 6288  df-nqqs 6332 This theorem is referenced by:  ltexprlemm  6572  ltexprlemopl  6573  ltexprlemlol  6574  ltexprlemdisj  6578  ltexprlemloc  6579  ltexprlemfl  6581  ltexprlemrl  6582
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