Theorem divap1d 8980

Description: If two complex numbers are apart, their quotient is apart from one. (Contributed by Jim Kingdon, 20-Mar-2020.)

Hypotheses

Ref	Expression
divcld.1	|- ( ph -> A e. CC )
divcld.2	|- ( ph -> B e. CC )
divclapd.3	|- ( ph -> B # 0 )
divap1d.4	|- ( ph -> A # B )

Assertion

Ref	Expression
divap1d	|- ( ph -> ( A / B ) # 1 )

Proof of Theorem divap1d

Step	Hyp	Ref	Expression
1		divap1d.4	. . 3 |- ( ph -> A # B )
2		divcld.1	. . . 4 |- ( ph -> A e. CC )
3		divcld.2	. . . 4 |- ( ph -> B e. CC )
4		divclapd.3	. . . . 5 |- ( ph -> B # 0 )
5	3, 4	recclapd 8960	. . . 4 |- ( ph -> ( 1 / B ) e. CC )
6	3, 4	recap0d 8961	. . . 4 |- ( ph -> ( 1 / B ) # 0 )
7		apmul1 8967	. . . 4 |- ( ( A e. CC /\ B e. CC /\ ( ( 1 / B ) e. CC /\ ( 1 / B ) # 0 ) ) -> ( A # B <-> ( A x. ( 1 / B ) ) # ( B x. ( 1 / B ) ) ) )
8	2, 3, 5, 6, 7	syl112anc 1277	. . 3 |- ( ph -> ( A # B <-> ( A x. ( 1 / B ) ) # ( B x. ( 1 / B ) ) ) )
9	1, 8	mpbid 147	. 2 |- ( ph -> ( A x. ( 1 / B ) ) # ( B x. ( 1 / B ) ) )
10	2, 3, 4	divrecapd 8972	. . 3 |- ( ph -> ( A / B ) = ( A x. ( 1 / B ) ) )
11	10	eqcomd 2237	. 2 |- ( ph -> ( A x. ( 1 / B ) ) = ( A / B ) )
12	3, 4	recidapd 8962	. 2 |- ( ph -> ( B x. ( 1 / B ) ) = 1 )
13	9, 11, 12	3brtr3d 4119	1 |- ( ph -> ( A / B ) # 1 )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   x. cmul 8036   # cap 8760   / cdiv 8851

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-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

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-rmo 2518  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-po 4393  df-iso 4394  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-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852

This theorem is referenced by: (None)