Theorem climuni 7052

Description: An infinite sequence of complex numbers converges to at most one limit.

Hypotheses

Ref	Expression
climuni.1	|- A e. V
climuni.2	|- B e. V

Assertion

Ref	Expression
climuni	|- ((F ~~> A /\ F ~~> B) -> A = B)

Proof of Theorem climuni

Step	Hyp	Ref	Expression
1		eqeq1 1479	. 2 |- (A = if((F ~~> A /\ F ~~> B), A, 0) -> (A = B <-> if((F ~~> A /\ F ~~> B), A, 0) = B))
2		eqeq2 1482	. 2 |- (B = if((F ~~> A /\ F ~~> B), B, 0) -> (if((F ~~> A /\ F ~~> B), A, 0) = B <-> if((F ~~> A /\ F ~~> B), A, 0) = if((F ~~> A /\ F ~~> B), B, 0)))
3		climuni.1	. . . 4 |- A e. V
4		0cn 5311	. . . . 5 |- 0 e. CC
5	4	elisseti 1815	. . . 4 |- 0 e. V
6	3, 5	ifex 2397	. . 3 |- if((F ~~> A /\ F ~~> B), A, 0) e. V
7		climuni.2	. . . 4 |- B e. V
8	7, 5	ifex 2397	. . 3 |- if((F ~~> A /\ F ~~> B), B, 0) e. V
9		breq2 2619	. . . . 5 |- (A = if((F ~~> A /\ F ~~> B), A, 0) -> (F ~~> A <-> F ~~> if((F ~~> A /\ F ~~> B), A, 0)))
10	9	anbi1d 616	. . . 4 |- (A = if((F ~~> A /\ F ~~> B), A, 0) -> ((F ~~> A /\ F ~~> B) <-> (F ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ F ~~> B)))
11		breq2 2619	. . . . 5 |- (B = if((F ~~> A /\ F ~~> B), B, 0) -> (F ~~> B <-> F ~~> if((F ~~> A /\ F ~~> B), B, 0)))
12	11	anbi2d 615	. . . 4 |- (B = if((F ~~> A /\ F ~~> B), B, 0) -> ((F ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ F ~~> B) <-> (F ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ F ~~> if((F ~~> A /\ F ~~> B), B, 0))))
13		breq1 2618	. . . . 5 |- (F = if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) -> (F ~~> if((F ~~> A /\ F ~~> B), A, 0) <-> if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), A, 0)))
14		breq1 2618	. . . . 5 |- (F = if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) -> (F ~~> if((F ~~> A /\ F ~~> B), B, 0) <-> if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), B, 0)))
15	13, 14	anbi12d 627	. . . 4 |- (F = if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) -> ((F ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ F ~~> if((F ~~> A /\ F ~~> B), B, 0)) <-> (if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), B, 0))))
16		breq2 2619	. . . . 5 |- (0 = if((F ~~> A /\ F ~~> B), A, 0) -> ((ZZ X. {0}) ~~> 0 <-> (ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), A, 0)))
17	16	anbi1d 616	. . . 4 |- (0 = if((F ~~> A /\ F ~~> B), A, 0) -> (((ZZ X. {0}) ~~> 0 /\ (ZZ X. {0}) ~~> 0) <-> ((ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ (ZZ X. {0}) ~~> 0)))
18		breq2 2619	. . . . 5 |- (0 = if((F ~~> A /\ F ~~> B), B, 0) -> ((ZZ X. {0}) ~~> 0 <-> (ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), B, 0)))
19	18	anbi2d 615	. . . 4 |- (0 = if((F ~~> A /\ F ~~> B), B, 0) -> (((ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ (ZZ X. {0}) ~~> 0) <-> ((ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ (ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), B, 0))))
20		breq1 2618	. . . . 5 |- ((ZZ X. {0}) = if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) -> ((ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), A, 0) <-> if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), A, 0)))
21		breq1 2618	. . . . 5 |- ((ZZ X. {0}) = if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) -> ((ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), B, 0) <-> if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), B, 0)))
22	20, 21	anbi12d 627	. . . 4 |- ((ZZ X. {0}) = if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) -> (((ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ (ZZ X. {0}) ~~> if((F ~~> A /\ F ~~> B), B, 0)) <-> (if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), B, 0))))
23		clim0 7050	. . . . 5 |- (ZZ X. {0}) ~~> 0
24	23, 23	pm3.2i 285	. . . 4 |- ((ZZ X. {0}) ~~> 0 /\ (ZZ X. {0}) ~~> 0)
25	10, 12, 15, 17, 19, 22, 24	elimhyp3v 2389	. . 3 |- (if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), A, 0) /\ if((F ~~> A /\ F ~~> B), F, (ZZ X. {0})) ~~> if((F ~~> A /\ F ~~> B), B, 0))
26	6, 8, 25	climunii 7051	. 2 |- if((F ~~> A /\ F ~~> B), A, 0) = if((F ~~> A /\ F ~~> B), B, 0)
27	1, 2, 26	dedth2v 2381	1 |- ((F ~~> A /\ F ~~> B) -> A = B)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  Vcvv 1808  ifcif 2358  {csn 2406   class class class wbr 2615   X. cxp 3164  CCcc 5215  0cc0 5217  ZZcz 5281   ~~> cli 6927

This theorem is referenced by:  climeu 7053  occllem7 9134  projlem31 9171

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-sup 4557  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-n 5883  df-2 5927  df-n0 6057  df-z 6093  df-seq1 6258  df-uz 6363  df-exp 6514  df-sqr 6615  df-re 6697  df-im 6698  df-cj 6699  df-abs 6700  df-clim 6928

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