Theorem stadd3i 11820

Description: If the sum of 3 states is 3, then each state is 1.

Hypotheses

Ref	Expression
stle.1	|- A e. CH
stle.2	|- B e. CH
stm1add3.3	|- C e. CH

Assertion

Ref	Expression
stadd3i	|- (S e. States -> ((((S` A) + (S` B)) + (S` C)) = 3 -> (S` A) = 1))

Proof of Theorem stadd3i

Step	Hyp	Ref	Expression
1		stle.1	. . . . . 6 |- A e. CH
2		stcl 11788	. . . . . 6 |- (S e. States -> (A e. CH -> (S` A) e. RR))
3	1, 2	mpi 55	. . . . 5 |- (S e. States -> (S` A) e. RR)
4	3	recnd 6468	. . . 4 |- (S e. States -> (S` A) e. CC)
5		stle.2	. . . . . 6 |- B e. CH
6		stcl 11788	. . . . . 6 |- (S e. States -> (B e. CH -> (S` B) e. RR))
7	5, 6	mpi 55	. . . . 5 |- (S e. States -> (S` B) e. RR)
8	7	recnd 6468	. . . 4 |- (S e. States -> (S` B) e. CC)
9		stm1add3.3	. . . . . 6 |- C e. CH
10		stcl 11788	. . . . . 6 |- (S e. States -> (C e. CH -> (S` C) e. RR))
11	9, 10	mpi 55	. . . . 5 |- (S e. States -> (S` C) e. RR)
12	11	recnd 6468	. . . 4 |- (S e. States -> (S` C) e. CC)
13		addass 6460	. . . 4 |- (((S` A) e. CC /\ (S` B) e. CC /\ (S` C) e. CC) -> (((S` A) + (S` B)) + (S` C)) = ((S` A) + ((S` B) + (S` C))))
14	4, 8, 12, 13	syl111anc 1100	. . 3 |- (S e. States -> (((S` A) + (S` B)) + (S` C)) = ((S` A) + ((S` B) + (S` C))))
15	14	eqeq1d 1892	. 2 |- (S e. States -> ((((S` A) + (S` B)) + (S` C)) = 3 <-> ((S` A) + ((S` B) + (S` C))) = 3))
16		3re 7165	. . . . . 6 |- 3 e. RR
17		readdcl 6455	. . . . . . . . 9 |- (((S` B) e. RR /\ (S` C) e. RR) -> ((S` B) + (S` C)) e. RR)
18	7, 11, 17	syl11anc 524	. . . . . . . 8 |- (S e. States -> ((S` B) + (S` C)) e. RR)
19		readdcl 6455	. . . . . . . 8 |- (((S` A) e. RR /\ ((S` B) + (S` C)) e. RR) -> ((S` A) + ((S` B) + (S` C))) e. RR)
20	3, 18, 19	syl11anc 524	. . . . . . 7 |- (S e. States -> ((S` A) + ((S` B) + (S` C))) e. RR)
21		ltne 6686	. . . . . . . 8 |- ((((S` A) + ((S` B) + (S` C))) e. RR /\ 3 e. RR /\ ((S` A) + ((S` B) + (S` C))) < 3) -> 3 =/= ((S` A) + ((S` B) + (S` C))))
22	21	3exp 1066	. . . . . . 7 |- (((S` A) + ((S` B) + (S` C))) e. RR -> (3 e. RR -> (((S` A) + ((S` B) + (S` C))) < 3 -> 3 =/= ((S` A) + ((S` B) + (S` C))))))
23	20, 22	syl 12	. . . . . 6 |- (S e. States -> (3 e. RR -> (((S` A) + ((S` B) + (S` C))) < 3 -> 3 =/= ((S` A) + ((S` B) + (S` C))))))
24	16, 23	mpi 55	. . . . 5 |- (S e. States -> (((S` A) + ((S` B) + (S` C))) < 3 -> 3 =/= ((S` A) + ((S` B) + (S` C)))))
25	24	necon2bd 2057	. . . 4 |- (S e. States -> (3 = ((S` A) + ((S` B) + (S` C))) -> -. ((S` A) + ((S` B) + (S` C))) < 3))
26		eqcom 1886	. . . 4 |- (((S` A) + ((S` B) + (S` C))) = 3 <-> 3 = ((S` A) + ((S` B) + (S` C))))
27	25, 26	syl5ib 223	. . 3 |- (S e. States -> (((S` A) + ((S` B) + (S` C))) = 3 -> -. ((S` A) + ((S` B) + (S` C))) < 3))
28		1re 6598	. . . . . . . . . . . 12 |- 1 e. RR
29	7, 28	jctir 317	. . . . . . . . . . 11 |- (S e. States -> ((S` B) e. RR /\ 1 e. RR))
30		readdcl 6455	. . . . . . . . . . 11 |- (((S` B) e. RR /\ 1 e. RR) -> ((S` B) + 1) e. RR)
31	29, 30	syl 12	. . . . . . . . . 10 |- (S e. States -> ((S` B) + 1) e. RR)
32	28, 28	readdcli 6487	. . . . . . . . . . 11 |- (1 + 1) e. RR
33	32	a1i 8	. . . . . . . . . 10 |- (S e. States -> (1 + 1) e. RR)
34		stle1 11797	. . . . . . . . . . . 12 |- (S e. States -> (C e. CH -> (S` C) <_ 1))
35	9, 34	mpi 55	. . . . . . . . . . 11 |- (S e. States -> (S` C) <_ 1)
36	28	a1i 8	. . . . . . . . . . . 12 |- (S e. States -> 1 e. RR)
37		leadd2 6809	. . . . . . . . . . . 12 |- (((S` C) e. RR /\ 1 e. RR /\ (S` B) e. RR) -> ((S` C) <_ 1 <-> ((S` B) + (S` C)) <_ ((S` B) + 1)))
38	11, 36, 7, 37	syl111anc 1100	. . . . . . . . . . 11 |- (S e. States -> ((S` C) <_ 1 <-> ((S` B) + (S` C)) <_ ((S` B) + 1)))
39	35, 38	mpbid 212	. . . . . . . . . 10 |- (S e. States -> ((S` B) + (S` C)) <_ ((S` B) + 1))
40		stle1 11797	. . . . . . . . . . . 12 |- (S e. States -> (B e. CH -> (S` B) <_ 1))
41	5, 40	mpi 55	. . . . . . . . . . 11 |- (S e. States -> (S` B) <_ 1)
42		leadd1 6808	. . . . . . . . . . . 12 |- (((S` B) e. RR /\ 1 e. RR /\ 1 e. RR) -> ((S` B) <_ 1 <-> ((S` B) + 1) <_ (1 + 1)))
43	7, 36, 36, 42	syl111anc 1100	. . . . . . . . . . 11 |- (S e. States -> ((S` B) <_ 1 <-> ((S` B) + 1) <_ (1 + 1)))
44	41, 43	mpbid 212	. . . . . . . . . 10 |- (S e. States -> ((S` B) + 1) <_ (1 + 1))
45	18, 31, 33, 39, 44	letrd 6696	. . . . . . . . 9 |- (S e. States -> ((S` B) + (S` C)) <_ (1 + 1))
46		leadd2 6809	. . . . . . . . . 10 |- ((((S` B) + (S` C)) e. RR /\ (1 + 1) e. RR /\ (S` A) e. RR) -> (((S` B) + (S` C)) <_ (1 + 1) <-> ((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1))))
47	18, 33, 3, 46	syl111anc 1100	. . . . . . . . 9 |- (S e. States -> (((S` B) + (S` C)) <_ (1 + 1) <-> ((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1))))
48	45, 47	mpbid 212	. . . . . . . 8 |- (S e. States -> ((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1)))
49	48	adantr 425	. . . . . . 7 |- ((S e. States /\ (S` A) < 1) -> ((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1)))
50		ltadd1 6806	. . . . . . . . . 10 |- (((S` A) e. RR /\ 1 e. RR /\ (1 + 1) e. RR) -> ((S` A) < 1 <-> ((S` A) + (1 + 1)) < (1 + (1 + 1))))
51	50	biimpd 170	. . . . . . . . 9 |- (((S` A) e. RR /\ 1 e. RR /\ (1 + 1) e. RR) -> ((S` A) < 1 -> ((S` A) + (1 + 1)) < (1 + (1 + 1))))
52	3, 36, 33, 51	syl111anc 1100	. . . . . . . 8 |- (S e. States -> ((S` A) < 1 -> ((S` A) + (1 + 1)) < (1 + (1 + 1))))
53	52	imp 377	. . . . . . 7 |- ((S e. States /\ (S` A) < 1) -> ((S` A) + (1 + 1)) < (1 + (1 + 1)))
54	3, 32	jctir 317	. . . . . . . . . 10 |- (S e. States -> ((S` A) e. RR /\ (1 + 1) e. RR))
55		readdcl 6455	. . . . . . . . . 10 |- (((S` A) e. RR /\ (1 + 1) e. RR) -> ((S` A) + (1 + 1)) e. RR)
56	54, 55	syl 12	. . . . . . . . 9 |- (S e. States -> ((S` A) + (1 + 1)) e. RR)
57	28, 32	readdcli 6487	. . . . . . . . . 10 |- (1 + (1 + 1)) e. RR
58	57	a1i 8	. . . . . . . . 9 |- (S e. States -> (1 + (1 + 1)) e. RR)
59		lelttr 6693	. . . . . . . . 9 |- ((((S` A) + ((S` B) + (S` C))) e. RR /\ ((S` A) + (1 + 1)) e. RR /\ (1 + (1 + 1)) e. RR) -> ((((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1)) /\ ((S` A) + (1 + 1)) < (1 + (1 + 1))) -> ((S` A) + ((S` B) + (S` C))) < (1 + (1 + 1))))
60	20, 56, 58, 59	syl111anc 1100	. . . . . . . 8 |- (S e. States -> ((((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1)) /\ ((S` A) + (1 + 1)) < (1 + (1 + 1))) -> ((S` A) + ((S` B) + (S` C))) < (1 + (1 + 1))))
61	60	adantr 425	. . . . . . 7 |- ((S e. States /\ (S` A) < 1) -> ((((S` A) + ((S` B) + (S` C))) <_ ((S` A) + (1 + 1)) /\ ((S` A) + (1 + 1)) < (1 + (1 + 1))) -> ((S` A) + ((S` B) + (S` C))) < (1 + (1 + 1))))
62	49, 53, 61	mp2and 767	. . . . . 6 |- ((S e. States /\ (S` A) < 1) -> ((S` A) + ((S` B) + (S` C))) < (1 + (1 + 1)))
63		df-3 7155	. . . . . . 7 |- 3 = (2 + 1)
64		df-2 7154	. . . . . . . 8 |- 2 = (1 + 1)
65	64	opreq1i 4892	. . . . . . 7 |- (2 + 1) = ((1 + 1) + 1)
66		ax1cn 6422	. . . . . . . 8 |- 1 e. CC
67	66, 66, 66	addassi 6477	. . . . . . 7 |- ((1 + 1) + 1) = (1 + (1 + 1))
68	63, 65, 67	3eqtrri 1913	. . . . . 6 |- (1 + (1 + 1)) = 3
69	62, 68	syl6breq 3376	. . . . 5 |- ((S e. States /\ (S` A) < 1) -> ((S` A) + ((S` B) + (S` C))) < 3)
70	69	ex 402	. . . 4 |- (S e. States -> ((S` A) < 1 -> ((S` A) + ((S` B) + (S` C))) < 3))
71	70	con3d 111	. . 3 |- (S e. States -> (-. ((S` A) + ((S` B) + (S` C))) < 3 -> -. (S` A) < 1))
72		stle1 11797	. . . . . 6 |- (S e. States -> (A e. CH -> (S` A) <_ 1))
73	1, 72	mpi 55	. . . . 5 |- (S e. States -> (S` A) <_ 1)
74	3, 28	jctir 317	. . . . . 6 |- (S e. States -> ((S` A) e. RR /\ 1 e. RR))
75		leloe 6688	. . . . . 6 |- (((S` A) e. RR /\ 1 e. RR) -> ((S` A) <_ 1 <-> ((S` A) < 1 \/ (S` A) = 1)))
76	74, 75	syl 12	. . . . 5 |- (S e. States -> ((S` A) <_ 1 <-> ((S` A) < 1 \/ (S` A) = 1)))
77	73, 76	mpbid 212	. . . 4 |- (S e. States -> ((S` A) < 1 \/ (S` A) = 1))
78	77	ord 249	. . 3 |- (S e. States -> (-. (S` A) < 1 -> (S` A) = 1))
79	27, 71, 78	3syld 31	. 2 |- (S e. States -> (((S` A) + ((S` B) + (S` C))) = 3 -> (S` A) = 1))
80	15, 79	sylbid 220	1 |- (S e. States -> ((((S` A) + (S` B)) + (S` C)) = 3 -> (S` A) = 1))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389   <_ cle 6448   < clt 6653  2c2 7145  3c3 7146  CHcch 10430  Statescst 10463

This theorem is referenced by:  golem2 11844

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-2 7154  df-3 7155  df-sh 10709  df-ch 10725  df-st 11784

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