Theorem efaddlem26 7363

Description: Lemma for efadd 7366. Show that the sequence of partial sum products H converges to the product of exponentiations. The key theorem used is climmul 7128.

Hypotheses

Ref	Expression
efaddlem26.1	|- A e. CC
efaddlem26.2	|- B e. CC
efaddlem26.3	|- F = {<.n, y>. | (n e. NN0 /\ y = sum_j e. (0...n)((A^j) / (!` j)))}
efaddlem26.4	|- G = {<.n, y>. | (n e. NN0 /\ y = sum_k e. (0...n)((B^k) / (!` k)))}
efaddlem26.5	|- H = {<.n, y>. | (n e. NN0 /\ y = (sum_j e. (0...n)((A^j) / (!` j)) x. sum_k e. (0...n)((B^k) / (!` k))))}

Assertion

Ref	Expression
efaddlem26	|- H ~~> ((exp` A) x. (exp` B))

Distinct variable groups:   j,n,y,A   k,n,y,B   j,k

Proof of Theorem efaddlem26

Step	Hyp	Ref	Expression
1		efaddlem26.1	. . . 4 |- A e. CC
2		efaddlem26.3	. . . . 5 |- F = {<.n, y>. | (n e. NN0 /\ y = sum_j e. (0...n)((A^j) / (!` j)))}
3	2	efcvgfsum 7315	. . . 4 |- (A e. CC -> F ~~> (exp` A))
4	1, 3	ax-mp 7	. . 3 |- F ~~> (exp` A)
5		efaddlem26.2	. . . 4 |- B e. CC
6		efaddlem26.4	. . . . 5 |- G = {<.n, y>. | (n e. NN0 /\ y = sum_k e. (0...n)((B^k) / (!` k)))}
7	6	efcvgfsum 7315	. . . 4 |- (B e. CC -> G ~~> (exp` B))
8	5, 7	ax-mp 7	. . 3 |- G ~~> (exp` B)
9	4, 8	pm3.2i 285	. 2 |- (F ~~> (exp` A) /\ G ~~> (exp` B))
10		0z 6146	. . 3 |- 0 e. ZZ
11		elnn0uz 6441	. . . . 5 |- (m e. NN0 <-> m e. (ZZ>` 0))
12		opreq2 3969	. . . . . . . . 9 |- (n = m -> (0...n) = (0...m))
13	12	sumeq1d 6990	. . . . . . . 8 |- (n = m -> sum_j e. (0...n)((A^j) / (!` j)) = sum_j e. (0...m)((A^j) / (!` j)))
14		sumex 6981	. . . . . . . 8 |- sum_j e. (0...m)((A^j) / (!` j)) e. V
15	13, 2, 14	fvopab4 3780	. . . . . . 7 |- (m e. NN0 -> (F` m) = sum_j e. (0...m)((A^j) / (!` j)))
16		elfznn0t 6496	. . . . . . . . . . 11 |- (j e. (0...m) -> j e. NN0)
17		eftclt 7303	. . . . . . . . . . . 12 |- ((A e. CC /\ j e. NN0) -> ((A^j) / (!` j)) e. CC)
18	1, 17	mpan 695	. . . . . . . . . . 11 |- (j e. NN0 -> ((A^j) / (!` j)) e. CC)
19	16, 18	syl 10	. . . . . . . . . 10 |- (j e. (0...m) -> ((A^j) / (!` j)) e. CC)
20	19	rgen 1698	. . . . . . . . 9 |- A.j e. (0...m)((A^j) / (!` j)) e. CC
21		fsumclt 7015	. . . . . . . . 9 |- ((m e. (ZZ>` 0) /\ A.j e. (0...m)((A^j) / (!` j)) e. CC) -> sum_j e. (0...m)((A^j) / (!` j)) e. CC)
22	20, 21	mpan2 696	. . . . . . . 8 |- (m e. (ZZ>` 0) -> sum_j e. (0...m)((A^j) / (!` j)) e. CC)
23	11, 22	sylbi 199	. . . . . . 7 |- (m e. NN0 -> sum_j e. (0...m)((A^j) / (!` j)) e. CC)
24	15, 23	eqeltrd 1548	. . . . . 6 |- (m e. NN0 -> (F` m) e. CC)
25	12	sumeq1d 6990	. . . . . . . 8 |- (n = m -> sum_k e. (0...n)((B^k) / (!` k)) = sum_k e. (0...m)((B^k) / (!` k)))
26		sumex 6981	. . . . . . . 8 |- sum_k e. (0...m)((B^k) / (!` k)) e. V
27	25, 6, 26	fvopab4 3780	. . . . . . 7 |- (m e. NN0 -> (G` m) = sum_k e. (0...m)((B^k) / (!` k)))
28		elfznn0t 6496	. . . . . . . . . . 11 |- (k e. (0...m) -> k e. NN0)
29		eftclt 7303	. . . . . . . . . . . 12 |- ((B e. CC /\ k e. NN0) -> ((B^k) / (!` k)) e. CC)
30	5, 29	mpan 695	. . . . . . . . . . 11 |- (k e. NN0 -> ((B^k) / (!` k)) e. CC)
31	28, 30	syl 10	. . . . . . . . . 10 |- (k e. (0...m) -> ((B^k) / (!` k)) e. CC)
32	31	rgen 1698	. . . . . . . . 9 |- A.k e. (0...m)((B^k) / (!` k)) e. CC
33		fsumclt 7015	. . . . . . . . 9 |- ((m e. (ZZ>` 0) /\ A.k e. (0...m)((B^k) / (!` k)) e. CC) -> sum_k e. (0...m)((B^k) / (!` k)) e. CC)
34	32, 33	mpan2 696	. . . . . . . 8 |- (m e. (ZZ>` 0) -> sum_k e. (0...m)((B^k) / (!` k)) e. CC)
35	11, 34	sylbi 199	. . . . . . 7 |- (m e. NN0 -> sum_k e. (0...m)((B^k) / (!` k)) e. CC)
36	27, 35	eqeltrd 1548	. . . . . 6 |- (m e. NN0 -> (G` m) e. CC)
37	13, 25	opreq12d 3978	. . . . . . . 8 |- (n = m -> (sum_j e. (0...n)((A^j) / (!` j)) x. sum_k e. (0...n)((B^k) / (!` k))) = (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))))
38		efaddlem26.5	. . . . . . . 8 |- H = {<.n, y>. | (n e. NN0 /\ y = (sum_j e. (0...n)((A^j) / (!` j)) x. sum_k e. (0...n)((B^k) / (!` k))))}
39		oprex 3983	. . . . . . . 8 |- (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))) e. V
40	37, 38, 39	fvopab4 3780	. . . . . . 7 |- (m e. NN0 -> (H` m) = (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))))
41	15, 27	opreq12d 3978	. . . . . . 7 |- (m e. NN0 -> ((F` m) x. (G` m)) = (sum_j e. (0...m)((A^j) / (!` j)) x. sum_k e. (0...m)((B^k) / (!` k))))
42	40, 41	eqtr4d 1510	. . . . . 6 |- (m e. NN0 -> (H` m) = ((F` m) x. (G` m)))
43	24, 36, 42	3jca 819	. . . . 5 |- (m e. NN0 -> ((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m) x. (G` m))))
44	11, 43	sylbir 201	. . . 4 |- (m e. (ZZ>` 0) -> ((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m) x. (G` m))))
45	44	rgen 1698	. . 3 |- A.m e. (ZZ>` 0)((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m) x. (G` m)))
46	10, 45	pm3.2i 285	. 2 |- (0 e. ZZ /\ A.m e. (ZZ>` 0)((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m) x. (G` m))))
47		nn0ex 6105	. . . 4 |- NN0 e. V
48	47, 2	fopabex2 3612	. . 3 |- F e. V
49	47, 6	fopabex2 3612	. . 3 |- G e. V
50	47, 38	fopabex2 3612	. . 3 |- H e. V
51		fvex 3732	. . 3 |- (exp` A) e. V
52		fvex 3732	. . 3 |- (exp` B) e. V
53	48, 49, 50, 51, 52	climmul 7128	. 2 |- (((F ~~> (exp` A) /\ G ~~> (exp` B)) /\ (0 e. ZZ /\ A.m e. (ZZ>` 0)((F` m) e. CC /\ (G` m) e. CC /\ (H` m) = ((F` m) x. (G` m))))) -> H ~~> ((exp` A) x. (exp` B)))
54	9, 46, 53	mp2an 697	1 |- H ~~> ((exp` A) x. (exp` B))

Syntax hints:   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645   class class class wbr 2619  {copab 2666  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234   x. cmul 5239   / cdiv 5294  NN0cn0 5297  ZZcz 5298  ZZ>cuz 6417  ...cfz 6467  ^cexp 6568  !cfa 6931   ~~> cli 6974  sum_csu 6979  expce 7293

This theorem is referenced by:  efaddlem28 7365

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q