Theorem rec11ap 8616

Description: Reciprocal is one-to-one. (Contributed by Jim Kingdon, 25-Feb-2020.)

Assertion

Ref	Expression
rec11ap	|- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )

Proof of Theorem rec11ap

Step	Hyp	Ref	Expression
1		1cnd 7925	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> 1 e. CC )
2		recclap 8585	. . . 4 |- ( ( B e. CC /\ B # 0 ) -> ( 1 / B ) e. CC )
3	2	adantl 275	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( 1 / B ) e. CC )
4		simpl 108	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A e. CC /\ A # 0 ) )
5		divmulap 8581	. . 3 |- ( ( 1 e. CC /\ ( 1 / B ) e. CC /\ ( A e. CC /\ A # 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> ( A x. ( 1 / B ) ) = 1 ) )
6	1, 3, 4, 5	syl3anc 1233	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> ( A x. ( 1 / B ) ) = 1 ) )
7		simpll 524	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> A e. CC )
8		simprl 526	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> B e. CC )
9		simprr 527	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> B # 0 )
10		divrecap 8594	. . . 4 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
11	7, 8, 9, 10	syl3anc 1233	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( A / B ) = ( A x. ( 1 / B ) ) )
12	11	eqeq1d 2179	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( A / B ) = 1 <-> ( A x. ( 1 / B ) ) = 1 ) )
13		diveqap1 8611	. . 3 |- ( ( A e. CC /\ B e. CC /\ B # 0 ) -> ( ( A / B ) = 1 <-> A = B ) )
14	7, 8, 9, 13	syl3anc 1233	. 2 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( A / B ) = 1 <-> A = B ) )
15	6, 12, 14	3bitr2d 215	1 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ B # 0 ) ) -> ( ( 1 / A ) = ( 1 / B ) <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   class class class wbr 3987  (class class class)co 5851   CCcc 7761   0cc0 7763   1c1 7764   x. cmul 7768   # cap 8489   / cdiv 8578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-mulrcl 7862  ax-addcom 7863  ax-mulcom 7864  ax-addass 7865  ax-mulass 7866  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-1rid 7870  ax-0id 7871  ax-rnegex 7872  ax-precex 7873  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879  ax-pre-mulgt0 7880  ax-pre-mulext 7881

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-reap 8483  df-ap 8490  df-div 8579

This theorem is referenced by:  rec11api  8659  rec11apd  8717