Theorem subdit 5350

Description: Distribution of multiplication over subtraction. Theorem I.5 of [Apostol] p. 18.

Assertion

Ref	Expression
subdit	|- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B - C)) = ((A x. B) - (A x. C)))

Proof of Theorem subdit

Step	Hyp	Ref	Expression
1		axdistr 5202	. . . . 5 |- ((A e. CC /\ C e. CC /\ (B - C) e. CC) -> (A x. (C + (B - C))) = ((A x. C) + (A x. (B - C))))
2		3simp1 785	. . . . 5 |- ((A e. CC /\ B e. CC /\ C e. CC) -> A e. CC)
3		3simp3 787	. . . . 5 |- ((A e. CC /\ B e. CC /\ C e. CC) -> C e. CC)
4		subclt 5290	. . . . . 6 |- ((B e. CC /\ C e. CC) -> (B - C) e. CC)
5	4	3adant1 794	. . . . 5 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (B - C) e. CC)
6	1, 2, 3, 5	syl3anc 855	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (C + (B - C))) = ((A x. C) + (A x. (B - C))))
7		pncan3t 5300	. . . . . . 7 |- ((C e. CC /\ B e. CC) -> (C + (B - C)) = B)
8	7	ancoms 436	. . . . . 6 |- ((B e. CC /\ C e. CC) -> (C + (B - C)) = B)
9	8	3adant1 794	. . . . 5 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (C + (B - C)) = B)
10	9	opreq2d 3915	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (C + (B - C))) = (A x. B))
11	6, 10	eqtr3d 1485	. . 3 |- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. C) + (A x. (B - C))) = (A x. B))
12		subaddt 5298	. . . 4 |- (((A x. B) e. CC /\ (A x. C) e. CC /\ (A x. (B - C)) e. CC) -> (((A x. B) - (A x. C)) = (A x. (B - C)) <-> ((A x. C) + (A x. (B - C))) = (A x. B)))
13		axmulcl 5196	. . . . 5 |- ((A e. CC /\ B e. CC) -> (A x. B) e. CC)
14	13	3adant3 796	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. B) e. CC)
15		axmulcl 5196	. . . . 5 |- ((A e. CC /\ C e. CC) -> (A x. C) e. CC)
16	15	3adant2 795	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. C) e. CC)
17		axmulcl 5196	. . . . . 6 |- ((A e. CC /\ (B - C) e. CC) -> (A x. (B - C)) e. CC)
18	17, 4	sylan2 451	. . . . 5 |- ((A e. CC /\ (B e. CC /\ C e. CC)) -> (A x. (B - C)) e. CC)
19	18	3impb 826	. . . 4 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B - C)) e. CC)
20	12, 14, 16, 19	syl3anc 855	. . 3 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (((A x. B) - (A x. C)) = (A x. (B - C)) <-> ((A x. C) + (A x. (B - C))) = (A x. B)))
21	11, 20	mpbird 196	. 2 |- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A x. B) - (A x. C)) = (A x. (B - C)))
22	21	eqcomd 1456	1 |- ((A e. CC /\ B e. CC /\ C e. CC) -> (A x. (B - C)) = ((A x. B) - (A x. C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 772   = wceq 1099   e. wcel 1105  (class class class)co 3902  CCcc 5155   + caddc 5160   x. cmul 5162   - cmin 5215

This theorem is referenced by:  subdirt 5351  subdi 5352  recextlem1 5606  qbtwnre 6167  expubndt 6490  subsqt 6524  climmullem5 7011  geoser 7120  mulc1cncf 7165  cos01bndlem3 7364  ipval2 8226  minveclem27 8437  2wsms 8824

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-inf2 4549

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-pss 2026  df-nul 2252  df-if 2333  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-int 2502  df-iun 2536  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-id 2797  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915  df-lim 2916  df-suc 2917  df-om 3095  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-fv 3161  df-rdg 3871  df-opr 3904  df-oprab 3905  df-1st 4017  df-2nd 4018  df-1o 4071  df-oadd 4073  df-omul 4074  df-er 4199  df-ec 4201  df-qs 4204  df-ni 4923  df-pli 4924  df-mi 4925  df-lti 4926  df-plpq 4958  df-mpq 4959  df-enq 4960  df-nq 4961  df-plq 4962  df-mq 4963  df-rq 4964  df-ltq 4965  df-1q 4966  df-np 5009  df-1p 5010  df-plp 5011  df-mp 5012  df-ltp 5013  df-plpr 5087  df-mpr 5088  df-enr 5089  df-nr 5090  df-plr 5091  df-mr 5092  df-0r 5094  df-1r 5095  df-m1r 5096  df-c 5163  df-0 5164  df-1 5165  df-i 5166  df-r 5167  df-plus 5168  df-mul 5169  df-sub 5279

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