Theorem cnegextlem1 5345

Description: Lemma for cnegext 5348.

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 5283	. . . 4 |- (A e. RR -> E.x e. RR (A + x) = 0)
2	1	3ad2ant1 800	. . 3 |- ((A e. RR /\ B e. CC /\ C e. CC) -> E.x e. RR (A + x) = 0)
3		axaddass 5277	. . . . . . . . . . . . 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 858	. . . . . . . . . . . 12 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((x + A) + B) = (x + (A + B)))
9		axaddass 5277	. . . . . . . . . . . . 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 858	. . . . . . . . . . . 12 |- (((A e. CC /\ (B e. CC /\ C e. CC)) /\ x e. CC) -> ((x + A) + C) = (x + (A + C)))
13	8, 12	eqeq12d 1489	. . . . . . . . . . 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 3969	. . . . . . . . . . 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 5275	. . . . . . . . . . . . 13 |- ((A e. CC /\ x e. CC) -> (A + x) = (x + A))
18	17	eqeq1d 1483	. . . . . . . . . . . 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 3968	. . . . . . . . . . . . . . 15 |- ((x + A) = 0 -> ((x + A) + B) = (0 + B))
21		opreq1 3968	. . . . . . . . . . . . . . 15 |- ((x + A) = 0 -> ((x + A) + C) = (0 + C))
22	20, 21	eqeq12d 1489	. . . . . . . . . . . . . 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 5329	. . . . . . . . . . . . . . . 16 |- (B e. CC -> (0 + B) = B)
25		addid2t 5329	. . . . . . . . . . . . . . . 16 |- (C e. CC -> (0 + C) = C)
26	24, 25	eqeqan12d 1490	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . 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 5313	. . . . . . 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 1747	. . . . 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 829	. . . 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 5313	. . . 4 |- (A e. RR -> A e. CC)
40	38, 39	syl3an1 859	. . 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 3969	. 2 |- (B = C -> (A + B) = (A + C))
43	41, 42	impbid1 517	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 775   = wceq 956   e. wcel 958  E.wrex 1646  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234   + caddc 5237

This theorem is referenced by:  cnegextlem3 5347

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246

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