Theorem crut 4531

Description: The real representation of complex numbers is unique. Proposition 10-1.3 of [Gleason] p. 130.

Assertion

Ref	Expression
crut	|- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A + (B x. i)) = (C + (D x. i)) -> (A = C /\ B = D)))

Proof of Theorem crut

Step	Hyp	Ref	Expression
1		opreq1 3006	. . . 4 |- (A = if(A e. RR, A, 0) -> (A + (B x. i)) = (if(A e. RR, A, 0) + (B x. i)))
2	1	cleq1d 1109	. . 3 |- (A = if(A e. RR, A, 0) -> ((A + (B x. i)) = (C + (D x. i)) <-> (if(A e. RR, A, 0) + (B x. i)) = (C + (D x. i))))
3		cleq1 1107	. . . 4 |- (A = if(A e. RR, A, 0) -> (A = C <-> if(A e. RR, A, 0) = C))
4	3	anbi1d 469	. . 3 |- (A = if(A e. RR, A, 0) -> ((A = C /\ B = D) <-> (if(A e. RR, A, 0) = C /\ B = D)))
5	2, 4	imbi12d 474	. 2 |- (A = if(A e. RR, A, 0) -> (((A + (B x. i)) = (C + (D x. i)) -> (A = C /\ B = D)) <-> ((if(A e. RR, A, 0) + (B x. i)) = (C + (D x. i)) -> (if(A e. RR, A, 0) = C /\ B = D))))
6		opreq1 3006	. . . . 5 |- (B = if(B e. RR, B, 0) -> (B x. i) = (if(B e. RR, B, 0) x. i))
7	6	opreq2d 3013	. . . 4 |- (B = if(B e. RR, B, 0) -> (if(A e. RR, A, 0) + (B x. i)) = (if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)))
8	7	cleq1d 1109	. . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) + (B x. i)) = (C + (D x. i)) <-> (if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (C + (D x. i))))
9		cleq1 1107	. . . 4 |- (B = if(B e. RR, B, 0) -> (B = D <-> if(B e. RR, B, 0) = D))
10	9	anbi2d 468	. . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) = C /\ B = D) <-> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D)))
11	8, 10	imbi12d 474	. 2 |- (B = if(B e. RR, B, 0) -> (((if(A e. RR, A, 0) + (B x. i)) = (C + (D x. i)) -> (if(A e. RR, A, 0) = C /\ B = D)) <-> ((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (C + (D x. i)) -> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D))))
12		opreq1 3006	. . . 4 |- (C = if(C e. RR, C, 0) -> (C + (D x. i)) = (if(C e. RR, C, 0) + (D x. i)))
13	12	cleq2d 1112	. . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (C + (D x. i)) <-> (if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (D x. i))))
14		cleq2 1110	. . . 4 |- (C = if(C e. RR, C, 0) -> (if(A e. RR, A, 0) = C <-> if(A e. RR, A, 0) = if(C e. RR, C, 0)))
15	14	anbi1d 469	. . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D)))
16	13, 15	imbi12d 474	. 2 |- (C = if(C e. RR, C, 0) -> (((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (C + (D x. i)) -> (if(A e. RR, A, 0) = C /\ if(B e. RR, B, 0) = D)) <-> ((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (D x. i)) -> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D))))
17		opreq1 3006	. . . . 5 |- (D = if(D e. RR, D, 0) -> (D x. i) = (if(D e. RR, D, 0) x. i))
18	17	opreq2d 3013	. . . 4 |- (D = if(D e. RR, D, 0) -> (if(C e. RR, C, 0) + (D x. i)) = (if(C e. RR, C, 0) + (if(D e. RR, D, 0) x. i)))
19	18	cleq2d 1112	. . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (D x. i)) <-> (if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (if(D e. RR, D, 0) x. i))))
20		cleq2 1110	. . . 4 |- (D = if(D e. RR, D, 0) -> (if(B e. RR, B, 0) = D <-> if(B e. RR, B, 0) = if(D e. RR, D, 0)))
21	20	anbi2d 468	. . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D) <-> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0))))
22	19, 21	imbi12d 474	. 2 |- (D = if(D e. RR, D, 0) -> (((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (D x. i)) -> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = D)) <-> ((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (if(D e. RR, D, 0) x. i)) -> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0)))))
23		ax0re 4063	. . . 4 |- 0 e. RR
24	23	elimel 1793	. . 3 |- if(A e. RR, A, 0) e. RR
25	23	elimel 1793	. . 3 |- if(B e. RR, B, 0) e. RR
26	23	elimel 1793	. . 3 |- if(C e. RR, C, 0) e. RR
27	23	elimel 1793	. . 3 |- if(D e. RR, D, 0) e. RR
28	24, 25, 26, 27	cru 4529	. 2 |- ((if(A e. RR, A, 0) + (if(B e. RR, B, 0) x. i)) = (if(C e. RR, C, 0) + (if(D e. RR, D, 0) x. i)) -> (if(A e. RR, A, 0) = if(C e. RR, C, 0) /\ if(B e. RR, B, 0) = if(D e. RR, D, 0)))
29	5, 11, 16, 22, 28	dedth4h 1789	1 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A + (B x. i)) = (C + (D x. i)) -> (A = C /\ B = D)))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  ifcif 1776  (class class class)co 3001  RRcr 4027  0cc0 4028  ici 4030   + caddc 4031   x. cmulc 4032

This theorem is referenced by:  creur 4532  creui 4533  rimul 4534  replimt 4798

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202</