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Theorem neap0mkv 16673
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 8178 . . . 4 |- 0 e. RR
2 neeq2 2416 . . . . . 6 |- ( y = 0 -> ( x =/= y <-> x =/= 0 ) )
3 breq2 4092 . . . . . 6 |- ( y = 0 -> ( x # y <-> x # 0 ) )
42, 3imbi12d 234 . . . . 5 |- ( y = 0 -> ( ( x =/= y -> x # y ) <-> ( x =/= 0 -> x # 0 ) ) )
54rspcv 2906 . . . 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 2595 . 2 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> A. x e. RR ( x =/= 0 -> x # 0 ) )
8 neeq1 2415 . . . . 5 |- ( x = z -> ( x =/= 0 <-> z =/= 0 ) )
9 breq1 4091 . . . . 5 |- ( x = z -> ( x # 0 <-> z # 0 ) )
108, 9imbi12d 234 . . . 4 |- ( x = z -> ( ( x =/= 0 -> x # 0 ) <-> ( z =/= 0 -> z # 0 ) ) )
1110cbvralv 2767 . . 3 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) <-> A. z e. RR ( z =/= 0 -> z # 0 ) )
12 neeq1 2415 . . . . . . 7 |- ( z = ( x - y ) -> ( z =/= 0 <-> ( x - y ) =/= 0 ) )
13 breq1 4091 . . . . . . 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 8559 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
1914, 15, 18rspcdva 2915 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 -> ( x - y ) # 0 ) )
2016recnd 8207 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
2117recnd 8207 . . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
2220, 21subeq0ad 8499 . . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
2322necon3bid 2443 . . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 <-> x =/= y ) )
24 subap0 8822 . . . . . 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 2614 . . 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 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   - cmin 8349   # cap 8760
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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148
This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761
This theorem is referenced by:  ltlenmkv  16674
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