Theorem mulcan 5698

Description: Cancellation law for multiplication. Theorem I.7 of [Apostol] p. 18.

Hypotheses

Ref	Expression
mulcan.1	|- A e. CC
mulcan.2	|- B e. CC
mulcan.3	|- C e. CC
mulcan.4	|- C =/= 0

Assertion

Ref	Expression
mulcan	|- ((C x. A) = (C x. B) <-> A = B)

Proof of Theorem mulcan

Step	Hyp	Ref	Expression
1		mulcan.3	. . . 4 |- C e. CC
2		mulcan.4	. . . 4 |- C =/= 0
3	1, 2	recex 5697	. . 3 |- E.x e. CC (C x. x) = 1
4		mulcan.1	. . . . . . . . . 10 |- A e. CC
5		axmulass 5290	. . . . . . . . . 10 |- ((x e. CC /\ C e. CC /\ A e. CC) -> ((x x. C) x. A) = (x x. (C x. A)))
6	4, 5	mp3an3 907	. . . . . . . . 9 |- ((x e. CC /\ C e. CC) -> ((x x. C) x. A) = (x x. (C x. A)))
7		mulcan.2	. . . . . . . . . 10 |- B e. CC
8		axmulass 5290	. . . . . . . . . 10 |- ((x e. CC /\ C e. CC /\ B e. CC) -> ((x x. C) x. B) = (x x. (C x. B)))
9	7, 8	mp3an3 907	. . . . . . . . 9 |- ((x e. CC /\ C e. CC) -> ((x x. C) x. B) = (x x. (C x. B)))
10	6, 9	eqeq12d 1492	. . . . . . . 8 |- ((x e. CC /\ C e. CC) -> (((x x. C) x. A) = ((x x. C) x. B) <-> (x x. (C x. A)) = (x x. (C x. B))))
11	1, 10	mpan2 698	. . . . . . 7 |- (x e. CC -> (((x x. C) x. A) = ((x x. C) x. B) <-> (x x. (C x. A)) = (x x. (C x. B))))
12		opreq2 3975	. . . . . . 7 |- ((C x. A) = (C x. B) -> (x x. (C x. A)) = (x x. (C x. B)))
13	11, 12	syl5bir 210	. . . . . 6 |- (x e. CC -> ((C x. A) = (C x. B) -> ((x x. C) x. A) = ((x x. C) x. B)))
14	13	adantr 391	. . . . 5 |- ((x e. CC /\ (C x. x) = 1) -> ((C x. A) = (C x. B) -> ((x x. C) x. A) = ((x x. C) x. B)))
15		axmulcom 5288	. . . . . . . . 9 |- ((C e. CC /\ x e. CC) -> (C x. x) = (x x. C))
16	1, 15	mpan 697	. . . . . . . 8 |- (x e. CC -> (C x. x) = (x x. C))
17	16	eqeq1d 1486	. . . . . . 7 |- (x e. CC -> ((C x. x) = 1 <-> (x x. C) = 1))
18		opreq1 3974	. . . . . . . . 9 |- ((x x. C) = 1 -> ((x x. C) x. A) = (1 x. A))
19	4	mulid2 5345	. . . . . . . . 9 |- (1 x. A) = A
20	18, 19	syl6eq 1526	. . . . . . . 8 |- ((x x. C) = 1 -> ((x x. C) x. A) = A)
21		opreq1 3974	. . . . . . . . 9 |- ((x x. C) = 1 -> ((x x. C) x. B) = (1 x. B))
22	7	mulid2 5345	. . . . . . . . 9 |- (1 x. B) = B
23	21, 22	syl6eq 1526	. . . . . . . 8 |- ((x x. C) = 1 -> ((x x. C) x. B) = B)
24	20, 23	eqeq12d 1492	. . . . . . 7 |- ((x x. C) = 1 -> (((x x. C) x. A) = ((x x. C) x. B) <-> A = B))
25	17, 24	syl6bi 214	. . . . . 6 |- (x e. CC -> ((C x. x) = 1 -> (((x x. C) x. A) = ((x x. C) x. B) <-> A = B)))
26	25	imp 350	. . . . 5 |- ((x e. CC /\ (C x. x) = 1) -> (((x x. C) x. A) = ((x x. C) x. B) <-> A = B))
27	14, 26	sylibd 202	. . . 4 |- ((x e. CC /\ (C x. x) = 1) -> ((C x. A) = (C x. B) -> A = B))
28	27	r19.23aiva 1747	. . 3 |- (E.x e. CC (C x. x) = 1 -> ((C x. A) = (C x. B) -> A = B))
29	3, 28	ax-mp 7	. 2 |- ((C x. A) = (C x. B) -> A = B)
30		opreq2 3975	. 2 |- (A = B -> (C x. A) = (C x. B))
31	29, 30	impbi 157	1 |- ((C x. A) = (C x. B) <-> A = B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   =/= wne 1588  E.wrex 1649  (class class class)co 3969  CCcc 5244  0cc0 5246  1c1 5247   x. cmul 5251

This theorem is referenced by:  mulcant2 5700

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503

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