Theorem fsum0diag3 7203

Description: Two ways to express "the sum of A(j) x. B(k) where j + k <_ N."

Assertion

Ref	Expression
fsum0diag3	|- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_j e. (0...N)(A x. sum_k e. (0...(N - j))B))

Distinct variable groups:   A,k   B,j   j,k,N

Proof of Theorem fsum0diag3

Step	Hyp	Ref	Expression
1		fsummulc1 6979	. . . . . . . . 9 |- (((N - j) e. (ZZ>` 0) /\ A e. CC /\ A.k e. (0...(N - j))B e. CC) -> (A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B))
2		fznn0subt 6438	. . . . . . . . . . . 12 |- ((N e. NN0 /\ j e. (0...N)) -> (N - j) e. NN0)
3		nn0uz 6378	. . . . . . . . . . . 12 |- NN0 = (ZZ>` 0)
4	2, 3	syl6eleq 1555	. . . . . . . . . . 11 |- ((N e. NN0 /\ j e. (0...N)) -> (N - j) e. (ZZ>` 0))
5	4	adantr 389	. . . . . . . . . 10 |- (((N e. NN0 /\ j e. (0...N)) /\ A e. CC) -> (N - j) e. (ZZ>` 0))
6	5	adantllr 397	. . . . . . . . 9 |- ((((N e. NN0 /\ A.k e. (0...N)B e. CC) /\ j e. (0...N)) /\ A e. CC) -> (N - j) e. (ZZ>` 0))
7		pm3.27 323	. . . . . . . . 9 |- ((((N e. NN0 /\ A.k e. (0...N)B e. CC) /\ j e. (0...N)) /\ A e. CC) -> A e. CC)
8		fsum0diaglem1 7199	. . . . . . . . . . . . . . 15 |- ((N e. NN0 /\ j e. (0...N)) -> (0...(N - j)) (_ (0...N))
9	8	sseld 2063	. . . . . . . . . . . . . 14 |- ((N e. NN0 /\ j e. (0...N)) -> (k e. (0...(N - j)) -> k e. (0...N)))
10	9	imim1d 28	. . . . . . . . . . . . 13 |- ((N e. NN0 /\ j e. (0...N)) -> ((k e. (0...N) -> B e. CC) -> (k e. (0...(N - j)) -> B e. CC)))
11	10	r19.20dv2 1708	. . . . . . . . . . . 12 |- ((N e. NN0 /\ j e. (0...N)) -> (A.k e. (0...N)B e. CC -> A.k e. (0...(N - j))B e. CC))
12	11	imp 350	. . . . . . . . . . 11 |- (((N e. NN0 /\ j e. (0...N)) /\ A.k e. (0...N)B e. CC) -> A.k e. (0...(N - j))B e. CC)
13	12	an1rs 489	. . . . . . . . . 10 |- (((N e. NN0 /\ A.k e. (0...N)B e. CC) /\ j e. (0...N)) -> A.k e. (0...(N - j))B e. CC)
14	13	adantr 389	. . . . . . . . 9 |- ((((N e. NN0 /\ A.k e. (0...N)B e. CC) /\ j e. (0...N)) /\ A e. CC) -> A.k e. (0...(N - j))B e. CC)
15	1, 6, 7, 14	syl3anc 857	. . . . . . . 8 |- ((((N e. NN0 /\ A.k e. (0...N)B e. CC) /\ j e. (0...N)) /\ A e. CC) -> (A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B))
16	15	ex 373	. . . . . . 7 |- (((N e. NN0 /\ A.k e. (0...N)B e. CC) /\ j e. (0...N)) -> (A e. CC -> (A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B)))
17	16	r19.20dva 1706	. . . . . 6 |- ((N e. NN0 /\ A.k e. (0...N)B e. CC) -> (A.j e. (0...N)A e. CC -> A.j e. (0...N)(A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B)))
18	17	ex 373	. . . . 5 |- (N e. NN0 -> (A.k e. (0...N)B e. CC -> (A.j e. (0...N)A e. CC -> A.j e. (0...N)(A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B))))
19	18	com23 32	. . . 4 |- (N e. NN0 -> (A.j e. (0...N)A e. CC -> (A.k e. (0...N)B e. CC -> A.j e. (0...N)(A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B))))
20	19	3imp 826	. . 3 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> A.j e. (0...N)(A x. sum_k e. (0...(N - j))B) = sum_k e. (0...(N - j))(A x. B))
21	20	sumeq2d 6937	. 2 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)(A x. sum_k e. (0...(N - j))B) = sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B))
22	21	eqcomd 1477	1 |- ((N e. NN0 /\ A.j e. (0...N)A e. CC /\ A.k e. (0...N)B e. CC) -> sum_j e. (0...N)sum_k e. (0...(N - j))(A x. B) = sum_j e. (0...N)(A x. sum_k e. (0...(N - j))B))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642  ` cfv 3177  (class class class)co 3954  CCcc 5212  0cc0 5214   x. cmul 5219   - cmin 5272  NN0cn0 5277  ZZ>cuz 6357  ...cfz 6407  sum_csu 6925

This theorem is referenced by:  fsum0diag4 7204

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-n 5881  df-n0 6055  df-z 6091  df-seq1 6253  df-shft 6286  df-uz 6358  df-fz 6408  df-seqz 6473  df-sum 6926

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