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Theorem neap0mkv 16437
Description: The analytic Markov principle can be expressed either with two arbitrary real numbers, or one arbitrary number and zero. (Contributed by Jim Kingdon, 23-Feb-2025.)
Assertion
Ref Expression
neap0mkv |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) <-> A. x e. RR ( x =/= 0 -> x # 0 ) )
Distinct variable group:   x, y
Proof of Theorem neap0mkv
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 0re 8146 . . . 4 |- 0 e. RR
2 neeq2 2414 . . . . . 6 |- ( y = 0 -> ( x =/= y <-> x =/= 0 ) )
3 breq2 4087 . . . . . 6 |- ( y = 0 -> ( x # y <-> x # 0 ) )
42, 3imbi12d 234 . . . . 5 |- ( y = 0 -> ( ( x =/= y -> x # y ) <-> ( x =/= 0 -> x # 0 ) ) )
54rspcv 2903 . . . 4 |- ( 0 e. RR -> ( A. y e. RR ( x =/= y -> x # y ) -> ( x =/= 0 -> x # 0 ) ) )
61, 5ax-mp 5 . . 3 |- ( A. y e. RR ( x =/= y -> x # y ) -> ( x =/= 0 -> x # 0 ) )
76ralimi 2593 . 2 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> A. x e. RR ( x =/= 0 -> x # 0 ) )
8 neeq1 2413 . . . . 5 |- ( x = z -> ( x =/= 0 <-> z =/= 0 ) )
9 breq1 4086 . . . . 5 |- ( x = z -> ( x # 0 <-> z # 0 ) )
108, 9imbi12d 234 . . . 4 |- ( x = z -> ( ( x =/= 0 -> x # 0 ) <-> ( z =/= 0 -> z # 0 ) ) )
1110cbvralv 2765 . . 3 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) <-> A. z e. RR ( z =/= 0 -> z # 0 ) )
12 neeq1 2413 . . . . . . 7 |- ( z = ( x - y ) -> ( z =/= 0 <-> ( x - y ) =/= 0 ) )
13 breq1 4086 . . . . . . 7 |- ( z = ( x - y ) -> ( z # 0 <-> ( x - y ) # 0 ) )
1412, 13imbi12d 234 . . . . . 6 |- ( z = ( x - y ) -> ( ( z =/= 0 -> z # 0 ) <-> ( ( x - y ) =/= 0 -> ( x - y ) # 0 ) ) )
15 simpl 109 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> A. z e. RR ( z =/= 0 -> z # 0 ) )
16 simprl 529 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
17 simprr 531 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
1816, 17resubcld 8527 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
1914, 15, 18rspcdva 2912 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 -> ( x - y ) # 0 ) )
2016recnd 8175 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
2117recnd 8175 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
2220, 21subeq0ad 8467 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
2322necon3bid 2441 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 <-> x =/= y ) )
24 subap0 8790 . . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( ( x - y ) # 0 <-> x # y ) )
2520, 21, 24syl2anc 411 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) # 0 <-> x # y ) )
2619, 23, 253imtr3d 202 . . . 4 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x =/= y -> x # y ) )
2726ralrimivva 2612 . . 3 |- ( A. z e. RR ( z =/= 0 -> z # 0 ) -> A. x e. RR A. y e. RR ( x =/= y -> x # y ) )
2811, 27sylbi 121 . 2 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) -> A. x e. RR A. y e. RR ( x =/= y -> x # y ) )
297, 28impbii 126 1 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) <-> A. x e. RR ( x =/= 0 -> x # 0 ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   class class class wbr 4083  (class class class)co 6001   CCcc 7997   RRcr 7998   0cc0 7999   - cmin 8317   # cap 8728
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116
This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729
This theorem is referenced by:  ltlenmkv  16438
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