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Theorem neap0mkv 16802
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 8239 . . . 4 |- 0 e. RR
2 neeq2 2417 . . . . . 6 |- ( y = 0 -> ( x =/= y <-> x =/= 0 ) )
3 breq2 4097 . . . . . 6 |- ( y = 0 -> ( x # y <-> x # 0 ) )
42, 3imbi12d 234 . . . . 5 |- ( y = 0 -> ( ( x =/= y -> x # y ) <-> ( x =/= 0 -> x # 0 ) ) )
54rspcv 2907 . . . 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 2596 . 2 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> A. x e. RR ( x =/= 0 -> x # 0 ) )
8 neeq1 2416 . . . . 5 |- ( x = z -> ( x =/= 0 <-> z =/= 0 ) )
9 breq1 4096 . . . . 5 |- ( x = z -> ( x # 0 <-> z # 0 ) )
108, 9imbi12d 234 . . . 4 |- ( x = z -> ( ( x =/= 0 -> x # 0 ) <-> ( z =/= 0 -> z # 0 ) ) )
1110cbvralv 2768 . . 3 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) <-> A. z e. RR ( z =/= 0 -> z # 0 ) )
12 neeq1 2416 . . . . . . 7 |- ( z = ( x - y ) -> ( z =/= 0 <-> ( x - y ) =/= 0 ) )
13 breq1 4096 . . . . . . 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 531 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. RR )
17 simprr 533 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. RR )
1816, 17resubcld 8619 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
1914, 15, 18rspcdva 2916 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 -> ( x - y ) # 0 ) )
2016recnd 8267 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
2117recnd 8267 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
2220, 21subeq0ad 8559 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
2322necon3bid 2444 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 <-> x =/= y ) )
24 subap0 8882 . . . . . 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 2615 . . 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 1398   e. wcel 2202   =/= wne 2403   A.wral 2511   class class class wbr 4093  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   - cmin 8409   # cap 8820
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821
This theorem is referenced by:  ltlenmkv  16803
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