Theorem div11t 5729

Description: One-to-one relationship for division.

Assertion

Ref	Expression
div11t	|- ((A e. CC /\ B e. CC /\ (C e. CC /\ C =/= 0)) -> ((A / C) = (B / C) <-> A = B))

Proof of Theorem div11t

Step	Hyp	Ref	Expression
1		opreq1 3959	. . . 4 |- (A = if(A e. CC, A, 1) -> (A / C) = (if(A e. CC, A, 1) / C))
2	1	eqeq1d 1480	. . 3 |- (A = if(A e. CC, A, 1) -> ((A / C) = (B / C) <-> (if(A e. CC, A, 1) / C) = (B / C)))
3		eqeq1 1478	. . 3 |- (A = if(A e. CC, A, 1) -> (A = B <-> if(A e. CC, A, 1) = B))
4	2, 3	bibi12d 628	. 2 |- (A = if(A e. CC, A, 1) -> (((A / C) = (B / C) <-> A = B) <-> ((if(A e. CC, A, 1) / C) = (B / C) <-> if(A e. CC, A, 1) = B)))
5		opreq1 3959	. . . 4 |- (B = if(B e. CC, B, 1) -> (B / C) = (if(B e. CC, B, 1) / C))
6	5	eqeq2d 1483	. . 3 |- (B = if(B e. CC, B, 1) -> ((if(A e. CC, A, 1) / C) = (B / C) <-> (if(A e. CC, A, 1) / C) = (if(B e. CC, B, 1) / C)))
7		eqeq2 1481	. . 3 |- (B = if(B e. CC, B, 1) -> (if(A e. CC, A, 1) = B <-> if(A e. CC, A, 1) = if(B e. CC, B, 1)))
8	6, 7	bibi12d 628	. 2 |- (B = if(B e. CC, B, 1) -> (((if(A e. CC, A, 1) / C) = (B / C) <-> if(A e. CC, A, 1) = B) <-> ((if(A e. CC, A, 1) / C) = (if(B e. CC, B, 1) / C) <-> if(A e. CC, A, 1) = if(B e. CC, B, 1))))
9		opreq2 3960	. . . 4 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> (if(A e. CC, A, 1) / C) = (if(A e. CC, A, 1) / if((C e. CC /\ C =/= 0), C, 1)))
10		opreq2 3960	. . . 4 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> (if(B e. CC, B, 1) / C) = (if(B e. CC, B, 1) / if((C e. CC /\ C =/= 0), C, 1)))
11	9, 10	eqeq12d 1486	. . 3 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> ((if(A e. CC, A, 1) / C) = (if(B e. CC, B, 1) / C) <-> (if(A e. CC, A, 1) / if((C e. CC /\ C =/= 0), C, 1)) = (if(B e. CC, B, 1) / if((C e. CC /\ C =/= 0), C, 1))))
12	11	bibi1d 618	. 2 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> (((if(A e. CC, A, 1) / C) = (if(B e. CC, B, 1) / C) <-> if(A e. CC, A, 1) = if(B e. CC, B, 1)) <-> ((if(A e. CC, A, 1) / if((C e. CC /\ C =/= 0), C, 1)) = (if(B e. CC, B, 1) / if((C e. CC /\ C =/= 0), C, 1)) <-> if(A e. CC, A, 1) = if(B e. CC, B, 1))))
13		ax1cn 5249	. . . 4 |- 1 e. CC
14	13	elimel 2390	. . 3 |- if(A e. CC, A, 1) e. CC
15	13	elimel 2390	. . 3 |- if(B e. CC, B, 1) e. CC
16		eleq1 1531	. . . . . 6 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> (C e. CC <-> if((C e. CC /\ C =/= 0), C, 1) e. CC))
17		neeq1 1587	. . . . . 6 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> (C =/= 0 <-> if((C e. CC /\ C =/= 0), C, 1) =/= 0))
18	16, 17	anbi12d 627	. . . . 5 |- (C = if((C e. CC /\ C =/= 0), C, 1) -> ((C e. CC /\ C =/= 0) <-> (if((C e. CC /\ C =/= 0), C, 1) e. CC /\ if((C e. CC /\ C =/= 0), C, 1) =/= 0)))
19		eleq1 1531	. . . . . 6 |- (1 = if((C e. CC /\ C =/= 0), C, 1) -> (1 e. CC <-> if((C e. CC /\ C =/= 0), C, 1) e. CC))
20		neeq1 1587	. . . . . 6 |- (1 = if((C e. CC /\ C =/= 0), C, 1) -> (1 =/= 0 <-> if((C e. CC /\ C =/= 0), C, 1) =/= 0))
21	19, 20	anbi12d 627	. . . . 5 |- (1 = if((C e. CC /\ C =/= 0), C, 1) -> ((1 e. CC /\ 1 =/= 0) <-> (if((C e. CC /\ C =/= 0), C, 1) e. CC /\ if((C e. CC /\ C =/= 0), C, 1) =/= 0)))
22		ax1ne0 5260	. . . . . 6 |- 1 =/= 0
23	13, 22	pm3.2i 285	. . . . 5 |- (1 e. CC /\ 1 =/= 0)
24	18, 21, 23	elimhyp 2386	. . . 4 |- (if((C e. CC /\ C =/= 0), C, 1) e. CC /\ if((C e. CC /\ C =/= 0), C, 1) =/= 0)
25	24	pm3.26i 320	. . 3 |- if((C e. CC /\ C =/= 0), C, 1) e. CC
26	24	pm3.27i 324	. . 3 |- if((C e. CC /\ C =/= 0), C, 1) =/= 0
27	14, 15, 25, 26	div11 5728	. 2 |- ((if(A e. CC, A, 1) / if((C e. CC /\ C =/= 0), C, 1)) = (if(B e. CC, B, 1) / if((C e. CC /\ C =/= 0), C, 1)) <-> if(A e. CC, A, 1) = if(B e. CC, B, 1))
28	4, 8, 12, 27	dedth3h 2384	1 |- ((A e. CC /\ B e. CC /\ (C e. CC /\ C =/= 0)) -> ((A / C) = (B / C) <-> A = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956   =/= wne 1582  ifcif 2357  (class class class)co 3954  CCcc 5212  0cc0 5214  1c1 5215   / cdiv 5274

This theorem is referenced by:  diveq0t 5732

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680

Copyright terms: Public domain