Theorem neap0mkv 16210

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 8107	. . . 4 |- 0 e. RR
2		neeq2 2392	. . . . . 6 |- ( y = 0 -> ( x =/= y <-> x =/= 0 ) )
3		breq2 4063	. . . . . 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 2880	. . . 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 2571	. 2 |- ( A. x e. RR A. y e. RR ( x =/= y -> x # y ) -> A. x e. RR ( x =/= 0 -> x # 0 ) )
8		neeq1 2391	. . . . 5 |- ( x = z -> ( x =/= 0 <-> z =/= 0 ) )
9		breq1 4062	. . . . 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 2742	. . 3 |- ( A. x e. RR ( x =/= 0 -> x # 0 ) <-> A. z e. RR ( z =/= 0 -> z # 0 ) )
12		neeq1 2391	. . . . . . 7 |- ( z = ( x - y ) -> ( z =/= 0 <-> ( x - y ) =/= 0 ) )
13		breq1 4062	. . . . . . 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 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 )
18	16, 17	resubcld 8488	. . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( x - y ) e. RR )
19	14, 15, 18	rspcdva 2889	. . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 -> ( x - y ) # 0 ) )
20	16	recnd 8136	. . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> x e. CC )
21	17	recnd 8136	. . . . . . 7 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> y e. CC )
22	20, 21	subeq0ad 8428	. . . . . 6 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) = 0 <-> x = y ) )
23	22	necon3bid 2419	. . . . 5 |- ( ( A. z e. RR ( z =/= 0 -> z # 0 ) /\ ( x e. RR /\ y e. RR ) ) -> ( ( x - y ) =/= 0 <-> x =/= y ) )
24		subap0 8751	. . . . . 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 2590	. . 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 1373   e. wcel 2178   =/= wne 2378   A.wral 2486   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   - cmin 8278   # cap 8689

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690

This theorem is referenced by:  ltlenmkv  16211