Theorem neap0mkv 16981

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.

Step	Hyp	Ref	Expression
1		0re 8290	. . . 4 |- 0 e. RR
2		neeq2 2428	. . . . . 6 |- ( y = 0 -> ( x =/= y <-> x =/= 0 ) )
3		breq2 4118	. . . . . 6 |- ( y = 0 -> ( x # y <-> x # 0 ) )
4	2, 3	imbi12d 234	. . . . 5 |- ( y = 0 -> ( ( x =/= y -> x # y ) <-> ( x =/= 0 -> x # 0 ) ) )
5	4	rspcv 2919	. . . 4 |- ( 0 e. RR -> ( A. y e. RR ( x =/= y -> x # y ) -> ( x =/= 0 -> x # 0 ) ) )
6	1, 5	ax-mp 5	. . 3 |- ( A. y e. RR ( x =/= y -> x # y ) -> ( x =/= 0 -> x # 0 ) )
7	6	ralimi 2607	. 2 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> A. x e. RR ( x =/= 0 -> x # 0 ) )
8		neeq1 2427	. . . . 5 |- ( x = z -> ( x =/= 0 <-> z =/= 0 ) )
9		breq1 4117	. . . . 5 |- ( x = z -> ( x # 0 <-> z # 0 ) )
10	8, 9	imbi12d 234	. . . 4 |- ( x = z -> ( ( x =/= 0 -> x # 0 ) <-> ( z =/= 0 -> z # 0 ) ) )
11	10	cbvralv 2780	. . 3 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) <-> A. z e. RR ( z =/= 0 -> z # 0 ) )
12		neeq1 2427	. . . . . . 7 |- ( z = ( x - y ) -> ( z =/= 0 <-> ( x - y ) =/= 0 ) )
13		breq1 4117	. . . . . . 7 |- ( z = ( x - y ) -> ( z # 0 <-> ( x - y ) # 0 ) )
14	12, 13	imbi12d 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 )
18	16, 17	resubcld 8671	. . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
19	14, 15, 18	rspcdva 2928	. . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 -> ( x - y ) # 0 ) )
20	16	recnd 8318	. . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
21	17	recnd 8318	. . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
22	20, 21	subeq0ad 8610	. . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
23	22	necon3bid 2455	. . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 <-> x =/= y ) )
24		subap0 8934	. . . . . 6 |- ( ( x e. CC /\ y e. CC ) -> ( ( x - y ) # 0 <-> x # y ) )
25	20, 21, 24	syl2anc 411	. . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) # 0 <-> x # y ) )
26	19, 23, 25	3imtr3d 202	. . . 4 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x =/= y -> x # y ) )
27	26	ralrimivva 2626	. . 3 |- ( A. z e. RR ( z =/= 0 -> z # 0 ) -> A. x e. RR A. y e. RR ( x =/= y -> x # y ) )
28	11, 27	sylbi 121	. 2 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) -> A. x e. RR A. y e. RR ( x =/= y -> x # y ) )
29	7, 28	impbii 126	1 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) <-> A. x e. RR ( x =/= 0 -> x # 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   class class class wbr 4114  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   - cmin 8460   # cap 8872

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873

This theorem is referenced by:  ltlenmkv  16982