Theorem fsumdivcALT 6989

Description: A finite sum divided by a constant. (An experiment: this version of fsumdivc 6988 adds 5 bytes and 233 bytes to the compressed and uncompressed proofs, but saves 540 bytes on the HTML page.)

Assertion

Ref	Expression
fsumdivcALT	|- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> (sum_k e. (M...N)A / C) = sum_k e. (M...N)(A / C))

Distinct variable groups:   C,k   k,M   k,N

Proof of Theorem fsumdivcALT

Step	Hyp	Ref	Expression
1		fsummulc2 6987	. . 3 |- ((N e. (ZZ>` M) /\ (1 / C) e. CC /\ A.k e. (M...N)A e. CC) -> (sum_k e. (M...N)A x. (1 / C)) = sum_k e. (M...N)(A x. (1 / C)))
2		simpll 412	. . 3 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> N e. (ZZ>` M))
3		recclt 5694	. . . 4 |- ((C e. CC /\ C =/= 0) -> (1 / C) e. CC)
4	3	ad2ant2lr 410	. . 3 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> (1 / C) e. CC)
5		simprr 415	. . 3 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> A.k e. (M...N)A e. CC)
6	1, 2, 4, 5	syl3anc 857	. 2 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> (sum_k e. (M...N)A x. (1 / C)) = sum_k e. (M...N)(A x. (1 / C)))
7		divrect 5712	. . . 4 |- ((sum_k e. (M...N)A e. CC /\ C e. CC /\ C =/= 0) -> (sum_k e. (M...N)A / C) = (sum_k e. (M...N)A x. (1 / C)))
8	7	3expb 833	. . 3 |- ((sum_k e. (M...N)A e. CC /\ (C e. CC /\ C =/= 0)) -> (sum_k e. (M...N)A / C) = (sum_k e. (M...N)A x. (1 / C)))
9		fsumclt 6968	. . . 4 |- ((N e. (ZZ>` M) /\ A.k e. (M...N)A e. CC) -> sum_k e. (M...N)A e. CC)
10	9	ad2ant2rl 411	. . 3 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> sum_k e. (M...N)A e. CC)
11		pm3.27 323	. . . 4 |- ((N e. (ZZ>` M) /\ C e. CC) -> C e. CC)
12		pm3.26 319	. . . 4 |- ((C =/= 0 /\ A.k e. (M...N)A e. CC) -> C =/= 0)
13	11, 12	anim12i 333	. . 3 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> (C e. CC /\ C =/= 0))
14	8, 10, 13	sylanc 471	. 2 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> (sum_k e. (M...N)A / C) = (sum_k e. (M...N)A x. (1 / C)))
15		divrect 5712	. . . . . . . . 9 |- ((A e. CC /\ C e. CC /\ C =/= 0) -> (A / C) = (A x. (1 / C)))
16	15	3expb 833	. . . . . . . 8 |- ((A e. CC /\ (C e. CC /\ C =/= 0)) -> (A / C) = (A x. (1 / C)))
17	16	expcom 374	. . . . . . 7 |- ((C e. CC /\ C =/= 0) -> (A e. CC -> (A / C) = (A x. (1 / C))))
18	17	r19.20sdv 1708	. . . . . 6 |- ((C e. CC /\ C =/= 0) -> (A.k e. (M...N)A e. CC -> A.k e. (M...N)(A / C) = (A x. (1 / C))))
19	18	imp 350	. . . . 5 |- (((C e. CC /\ C =/= 0) /\ A.k e. (M...N)A e. CC) -> A.k e. (M...N)(A / C) = (A x. (1 / C)))
20	19	anasss 440	. . . 4 |- ((C e. CC /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> A.k e. (M...N)(A / C) = (A x. (1 / C)))
21	20	sumeq2d 6944	. . 3 |- ((C e. CC /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> sum_k e. (M...N)(A / C) = sum_k e. (M...N)(A x. (1 / C)))
22	21	adantll 392	. 2 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> sum_k e. (M...N)(A / C) = sum_k e. (M...N)(A x. (1 / C)))
23	6, 14, 22	3eqtr4d 1515	1 |- (((N e. (ZZ>` M) /\ C e. CC) /\ (C =/= 0 /\ A.k e. (M...N)A e. CC)) -> (sum_k e. (M...N)A / C) = sum_k e. (M...N)(A / C))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957   =/= wne 1583  A.wral 1643  ` cfv 3178  (class class class)co 3958  CCcc 5215  0cc0 5217  1c1 5218   x. cmul 5222   / cdiv 5277  ZZ>cuz 6362  ...cfz 6412  sum_csu 6932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-inf2 4608

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-rdg 3927  df-opr 3960  df-oprab 3961  df-1st 4072  df-2nd 4073  df-1o 4126  df-oadd 4128  df-omul 4129  df-er 4254  df-ec 4256  df-qs 4259  df-en 4360  df-dom 4361  df-sdom 4362  df-ni 4983  df-pli 4984  df-mi 4985  df-lti 4986  df-plpq 5018  df-mpq 5019  df-enq 5020  df-nq 5021  df-plq 5022  df-mq 5023  df-rq 5024  df-ltq 5025  df-1q 5026  df-np 5069  df-1p 5070  df-plp 5071  df-mp 5072  df-ltp 5073  df-plpr 5147  df-mpr 5148  df-enr 5149  df-nr 5150  df-plr 5151  df-mr 5152  df-ltr 5153  df-0r 5154  df-1r 5155  df-m1r 5156  df-c 5223  df-0 5224  df-1 5225  df-i 5226  df-r 5227  df-plus 5228  df-mul 5229  df-lt 5230  df-sub 5339  df-neg 5341  df-pnf 5470  df-mnf 5471  df-xr 5472  df-ltxr 5473  df-le 5474  df-div 5682  df-n 5883  df-n0 6057  df-z 6093  df-seq1 6258  df-shft 6291  df-uz 6363  df-fz 6413  df-seqz 6478  df-sum 6933

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