Theorem cnegextlem1 5325

Description: Lemma for cnegext 5328.

Assertion

Ref	Expression
cnegextlem1	|- ((A e. RR /\ B e. CC /\ C e. CC) -> ((A + B) = (A + C) <-> B = C))

Proof of Theorem cnegextlem1

Step	Hyp	Ref	Expression
1		axrnegex 5263	. . . 4 |- (A e. RR -> E.x e. RR (A + x) = 0)
2	1	3ad2ant1 799	. . 3 |- ((A e. RR /\ B e. CC /\ C e. CC) -> E.x e. RR (A + x) = 0)
3		axaddass 5257	. . . . . . . . . . . . 13 |- ((x e. CC /\ A e. CC /\ B e. CC) -> ((x + A) + B) = (x + (A + B)))
4		pm3.27 323	. . . . . . . . . . . . 13 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> x e. CC)
5		simpll 412	. . . . . . . . . . . . 13 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> A e. CC)
6		simprl 414	. . . . . . . . . . . . . 14 |- ((A e. CC /\ (B e. CC /\ C e. CC)) -> B e. CC)
7	6	adantr 389	. . . . . . . . . . . . 13 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> B e. CC)
8	3, 4, 5, 7	syl3anc 857	. . . . . . . . . . . 12 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((x + A) + B) = (x + (A + B)))
9		axaddass 5257	. . . . . . . . . . . . 13 |- ((x e. CC /\ A e. CC /\ C e. CC) -> ((x + A) + C) = (x + (A + C)))
10		simprr 415	. . . . . . . . . . . . . 14 |- ((A e. CC /\ (B e. CC /\ C e. CC)) -> C e. CC)
11	10	adantr 389	. . . . . . . . . . . . 13 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> C e. CC)
12	9, 4, 5, 11	syl3anc 857	. . . . . . . . . . . 12 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((x + A) + C) = (x + (A + C)))
13	8, 12	eqeq12d 1486	. . . . . . . . . . 11 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> (((x + A) + B) = ((x + A) + C) <-> (x + (A + B)) = (x + (A + C))))
14		opreq2 3960	. . . . . . . . . . 11 |- ((A + B) = (A + C) -> (x + (A + B)) = (x + (A + C)))
15	13, 14	syl5bir 210	. . . . . . . . . 10 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((A + B) = (A + C) -> ((x + A) + B) = ((x + A) + C)))
16	15	adantr 389	. . . . . . . . 9 |- ((((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) /\ (A + x) = 0) -> ((A + B) = (A + C) -> ((x + A) + B) = ((x + A) + C)))
17		axaddcom 5255	. . . . . . . . . . . . 13 |- ((A e. CC /\ x e. CC) -> (A + x) = (x + A))
18	17	eqeq1d 1480	. . . . . . . . . . . 12 |- ((A e. CC /\ x e. CC) -> ((A + x) = 0 <-> (x + A) = 0))
19	18	adantlr 393	. . . . . . . . . . 11 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((A + x) = 0 <-> (x + A) = 0))
20		opreq1 3959	. . . . . . . . . . . . . . 15 |- ((x + A) = 0 -> ((x + A) + B) = (0 + B))
21		opreq1 3959	. . . . . . . . . . . . . . 15 |- ((x + A) = 0 -> ((x + A) + C) = (0 + C))
22	20, 21	eqeq12d 1486	. . . . . . . . . . . . . 14 |- ((x + A) = 0 -> (((x + A) + B) = ((x + A) + C) <-> (0 + B) = (0 + C)))
23	22	adantl 388	. . . . . . . . . . . . 13 |- ((((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) /\ (x + A) = 0) -> (((x + A) + B) = ((x + A) + C) <-> (0 + B) = (0 + C)))
24		addid2t 5309	. . . . . . . . . . . . . . . 16 |- (B e. CC -> (0 + B) = B)
25		addid2t 5309	. . . . . . . . . . . . . . . 16 |- (C e. CC -> (0 + C) = C)
26	24, 25	eqeqan12d 1487	. . . . . . . . . . . . . . 15 |- ((B e. CC /\ C e. CC) -> ((0 + B) = (0 + C) <-> B = C))
27	26	adantl 388	. . . . . . . . . . . . . 14 |- ((A e. CC /\ (B e. CC /\ C e. CC)) -> ((0 + B) = (0 + C) <-> B = C))
28	27	ad2antrr 404	. . . . . . . . . . . . 13 |- ((((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) /\ (x + A) = 0) -> ((0 + B) = (0 + C) <-> B = C))
29	23, 28	bitrd 527	. . . . . . . . . . . 12 |- ((((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) /\ (x + A) = 0) -> (((x + A) + B) = ((x + A) + C) <-> B = C))
30	29	ex 373	. . . . . . . . . . 11 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((x + A) = 0 -> (((x + A) + B) = ((x + A) + C) <-> B = C)))
31	19, 30	sylbid 203	. . . . . . . . . 10 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((A + x) = 0 -> (((x + A) + B) = ((x + A) + C) <-> B = C)))
32	31	imp 350	. . . . . . . . 9 |- ((((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) /\ (A + x) = 0) -> (((x + A) + B) = ((x + A) + C) <-> B = C))
33	16, 32	sylibd 202	. . . . . . . 8 |- ((((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) /\ (A + x) = 0) -> ((A + B) = (A + C) -> B = C))
34	33	ex 373	. . . . . . 7 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((A + x) = 0 -> ((A + B) = (A + C) -> B = C)))
35		recnt 5293	. . . . . . 7 |- (x e. RR -> x e. CC)
36	34, 35	sylan2 451	. . . . . 6 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. RR) -> ((A + x) = 0 -> ((A + B) = (A + C) -> B = C)))
37	36	r19.23adva 1744	. . . . 5 |- ((A e. CC /\ (B e. CC /\ C e. CC)) -> (E.x e. RR (A + x) = 0 -> ((A + B) = (A + C) -> B = C)))
38	37	3impb 828	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (E.x e. RR (A + x) = 0 -> ((A + B) = (A + C) -> B = C)))
39		recnt 5293	. . . 4 |- (A e. RR -> A e. CC)
40	38, 39	syl3an1 858	. . 3 |- ((A e. RR /\ B e. CC /\ C e. CC) -> (E.x e. RR (A + x) = 0 -> ((A + B) = (A + C) -> B = C)))
41	2, 40	mpd 26	. 2 |- ((A e. RR /\ B e. CC /\ C e. CC) -> ((A + B) = (A + C) -> B = C))
42		opreq2 3960	. 2 |- (B = C -> (A + B) = (A + C))
43	41, 42	impbid1 516	1 |- ((A e. RR /\ B e. CC /\ C e. CC) -> ((A + B) = (A + C) <-> B = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  E.wrex 1643  (class class class)co 3954  CCcc 5212  RRcr 5213  0cc0 5214   + caddc 5217

This theorem is referenced by:  cnegextlem3 5327

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-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-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-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-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

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