Theorem xrsmulgzz 29678

Description: The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)

Assertion

Ref	Expression
xrsmulgzz	|- ( ( A e. ZZ /\ B e. RR* ) -> ( A (.g ` RR*s ) B ) = ( A xe B ) )

Proof of Theorem xrsmulgzz

Dummy variables n m x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( n = 0 -> ( n (.g ` RR*s ) B ) = ( 0 (.g ` RR*s ) B ) )
2		oveq1 6657	. . . 4 |- ( n = 0 -> ( n xe B ) = ( 0 xe B ) )
3	1, 2	eqeq12d 2637	. . 3 |- ( n = 0 -> ( ( n (.g ` RR*s ) B ) = ( n xe B ) <-> ( 0 (.g ` RR*s ) B ) = ( 0 xe B ) ) )
4		oveq1 6657	. . . 4 |- ( n = m -> ( n (.g ` RR*s ) B ) = ( m (.g ` RR*s ) B ) )
5		oveq1 6657	. . . 4 |- ( n = m -> ( n xe B ) = ( m xe B ) )
6	4, 5	eqeq12d 2637	. . 3 |- ( n = m -> ( ( n (.g ` RR*s ) B ) = ( n xe B ) <-> ( m (.g ` RR*s ) B ) = ( m xe B ) ) )
7		oveq1 6657	. . . 4 |- ( n = ( m + 1 ) -> ( n (.g ` RR*s ) B ) = ( ( m + 1 ) (.g ` RR*s ) B ) )
8		oveq1 6657	. . . 4 |- ( n = ( m + 1 ) -> ( n xe B ) = ( ( m + 1 ) xe B ) )
9	7, 8	eqeq12d 2637	. . 3 |- ( n = ( m + 1 ) -> ( ( n (.g ` RR*s ) B ) = ( n xe B ) <-> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m + 1 ) xe B ) ) )
10		oveq1 6657	. . . 4 |- ( n = -u m -> ( n (.g ` RR*s ) B ) = ( -u m (.g ` RR*s ) B ) )
11		oveq1 6657	. . . 4 |- ( n = -u m -> ( n xe B ) = ( -u m xe B ) )
12	10, 11	eqeq12d 2637	. . 3 |- ( n = -u m -> ( ( n (.g ` RR*s ) B ) = ( n xe B ) <-> ( -u m (.g ` RR*s ) B ) = ( -u m xe B ) ) )
13		oveq1 6657	. . . 4 |- ( n = A -> ( n (.g ` RR*s ) B ) = ( A (.g ` RR*s ) B ) )
14		oveq1 6657	. . . 4 |- ( n = A -> ( n xe B ) = ( A xe B ) )
15	13, 14	eqeq12d 2637	. . 3 |- ( n = A -> ( ( n (.g ` RR*s ) B ) = ( n xe B ) <-> ( A (.g ` RR*s ) B ) = ( A xe B ) ) )
16		xrsbas 19762	. . . . 5 |- RR* = ( Base ` RR*s )
17		xrs0 29675	. . . . 5 |- 0 = ( 0g ` RR*s )
18		eqid 2622	. . . . 5 |- (.g ` RR*s ) = (.g ` RR*s )
19	16, 17, 18	mulg0 17546	. . . 4 |- ( B e. RR* -> ( 0 (.g ` RR*s ) B ) = 0 )
20		xmul02 12098	. . . 4 |- ( B e. RR* -> ( 0 xe B ) = 0 )
21	19, 20	eqtr4d 2659	. . 3 |- ( B e. RR* -> ( 0 (.g ` RR*s ) B ) = ( 0 xe B ) )
22		simpr 477	. . . . . 6 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( m (.g ` RR*s ) B ) = ( m xe B ) )
23	22	oveq1d 6665	. . . . 5 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m (.g ` RR*s ) B ) +e B ) = ( ( m xe B ) +e B ) )
24		simpr 477	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ m e. NN ) -> m e. NN )
25		simpll 790	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ m e. NN ) -> B e. RR* )
26		xrsadd 19763	. . . . . . . . 9 |- +e = ( +g ` RR*s )
27	16, 18, 26	mulgnnp1 17549	. . . . . . . 8 |- ( ( m e. NN /\ B e. RR* ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m (.g ` RR*s ) B ) +e B ) )
28	24, 25, 27	syl2anc 693	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ m e. NN ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m (.g ` RR*s ) B ) +e B ) )
29		simpr 477	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ m = 0 ) -> m = 0 )
30		simpll 790	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ m = 0 ) -> B e. RR* )
31		xaddid2 12073	. . . . . . . . . 10 |- ( B e. RR* -> ( 0 +e B ) = B )
32	31	adantl 482	. . . . . . . . 9 |- ( ( m = 0 /\ B e. RR* ) -> ( 0 +e B ) = B )
33		simpl 473	. . . . . . . . . . . 12 |- ( ( m = 0 /\ B e. RR* ) -> m = 0 )
34	33	oveq1d 6665	. . . . . . . . . . 11 |- ( ( m = 0 /\ B e. RR* ) -> ( m (.g ` RR*s ) B ) = ( 0 (.g ` RR*s ) B ) )
35	19	adantl 482	. . . . . . . . . . 11 |- ( ( m = 0 /\ B e. RR* ) -> ( 0 (.g ` RR*s ) B ) = 0 )
36	34, 35	eqtrd 2656	. . . . . . . . . 10 |- ( ( m = 0 /\ B e. RR* ) -> ( m (.g ` RR*s ) B ) = 0 )
37	36	oveq1d 6665	. . . . . . . . 9 |- ( ( m = 0 /\ B e. RR* ) -> ( ( m (.g ` RR*s ) B ) +e B ) = ( 0 +e B ) )
38	33	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( m = 0 /\ B e. RR* ) -> ( m + 1 ) = ( 0 + 1 ) )
39		0p1e1 11132	. . . . . . . . . . . 12 |- ( 0 + 1 ) = 1
40	38, 39	syl6eq 2672	. . . . . . . . . . 11 |- ( ( m = 0 /\ B e. RR* ) -> ( m + 1 ) = 1 )
41	40	oveq1d 6665	. . . . . . . . . 10 |- ( ( m = 0 /\ B e. RR* ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( 1 (.g ` RR*s ) B ) )
42	16, 18	mulg1 17548	. . . . . . . . . . 11 |- ( B e. RR* -> ( 1 (.g ` RR*s ) B ) = B )
43	42	adantl 482	. . . . . . . . . 10 |- ( ( m = 0 /\ B e. RR* ) -> ( 1 (.g ` RR*s ) B ) = B )
44	41, 43	eqtrd 2656	. . . . . . . . 9 |- ( ( m = 0 /\ B e. RR* ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = B )
45	32, 37, 44	3eqtr4rd 2667	. . . . . . . 8 |- ( ( m = 0 /\ B e. RR* ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m (.g ` RR*s ) B ) +e B ) )
46	29, 30, 45	syl2anc 693	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ m = 0 ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m (.g ` RR*s ) B ) +e B ) )
47		simpr 477	. . . . . . . 8 |- ( ( B e. RR* /\ m e. NN0 ) -> m e. NN0 )
48		elnn0 11294	. . . . . . . 8 |- ( m e. NN0 <-> ( m e. NN \/ m = 0 ) )
49	47, 48	sylib 208	. . . . . . 7 |- ( ( B e. RR* /\ m e. NN0 ) -> ( m e. NN \/ m = 0 ) )
50	28, 46, 49	mpjaodan 827	. . . . . 6 |- ( ( B e. RR* /\ m e. NN0 ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m (.g ` RR*s ) B ) +e B ) )
51	50	adantr 481	. . . . 5 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m (.g ` RR*s ) B ) +e B ) )
52		nn0ssre 11296	. . . . . . . . 9 |- NN0 C_ RR
53		ressxr 10083	. . . . . . . . 9 |- RR C_ RR*
54	52, 53	sstri 3612	. . . . . . . 8 |- NN0 C_ RR*
55	47	adantr 481	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> m e. NN0 )
56	54, 55	sseldi 3601	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> m e. RR* )
57		nn0ge0 11318	. . . . . . . 8 |- ( m e. NN0 -> 0 <_ m )
58	57	ad2antlr 763	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> 0 <_ m )
59		1re 10039	. . . . . . . . 9 |- 1 e. RR
60	59	rexri 10097	. . . . . . . 8 |- 1 e. RR*
61	60	a1i 11	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> 1 e. RR* )
62		0le1 10551	. . . . . . . 8 |- 0 <_ 1
63	62	a1i 11	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> 0 <_ 1 )
64		simpll 790	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> B e. RR* )
65		xadddi2r 12128	. . . . . . 7 |- ( ( ( m e. RR* /\ 0 <_ m ) /\ ( 1 e. RR* /\ 0 <_ 1 ) /\ B e. RR* ) -> ( ( m +e 1 ) xe B ) = ( ( m xe B ) +e ( 1 xe B ) ) )
66	56, 58, 61, 63, 64, 65	syl221anc 1337	. . . . . 6 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m +e 1 ) xe B ) = ( ( m xe B ) +e ( 1 xe B ) ) )
67	52, 55	sseldi 3601	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> m e. RR )
68	59	a1i 11	. . . . . . . 8 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> 1 e. RR )
69		rexadd 12063	. . . . . . . 8 |- ( ( m e. RR /\ 1 e. RR ) -> ( m +e 1 ) = ( m + 1 ) )
70	67, 68, 69	syl2anc 693	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( m +e 1 ) = ( m + 1 ) )
71	70	oveq1d 6665	. . . . . 6 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m +e 1 ) xe B ) = ( ( m + 1 ) xe B ) )
72		xmulid2 12110	. . . . . . . 8 |- ( B e. RR* -> ( 1 xe B ) = B )
73	64, 72	syl 17	. . . . . . 7 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( 1 xe B ) = B )
74	73	oveq2d 6666	. . . . . 6 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m xe B ) +e ( 1 xe B ) ) = ( ( m xe B ) +e B ) )
75	66, 71, 74	3eqtr3d 2664	. . . . 5 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m + 1 ) xe B ) = ( ( m xe B ) +e B ) )
76	23, 51, 75	3eqtr4d 2666	. . . 4 |- ( ( ( B e. RR* /\ m e. NN0 ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m + 1 ) xe B ) )
77	76	exp31 630	. . 3 |- ( B e. RR* -> ( m e. NN0 -> ( ( m (.g ` RR*s ) B ) = ( m xe B ) -> ( ( m + 1 ) (.g ` RR*s ) B ) = ( ( m + 1 ) xe B ) ) ) )
78		xnegeq 12038	. . . . . 6 |- ( ( m (.g ` RR*s ) B ) = ( m xe B ) -> -e ( m (.g ` RR*s ) B ) = -e ( m xe B ) )
79	78	adantl 482	. . . . 5 |- ( ( ( B e. RR* /\ m e. NN ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> -e ( m (.g ` RR*s ) B ) = -e ( m xe B ) )
80		eqid 2622	. . . . . . . . 9 |- ( invg ` RR*s ) = ( invg ` RR*s )
81	16, 18, 80	mulgnegnn 17551	. . . . . . . 8 |- ( ( m e. NN /\ B e. RR* ) -> ( -u m (.g ` RR*s ) B ) = ( ( invg ` RR*s ) ` ( m (.g ` RR*s ) B ) ) )
82	81	ancoms 469	. . . . . . 7 |- ( ( B e. RR* /\ m e. NN ) -> ( -u m (.g ` RR*s ) B ) = ( ( invg ` RR*s ) ` ( m (.g ` RR*s ) B ) ) )
83		xrsex 19761	. . . . . . . . . . . 12 |- RR*s e. _V
84	83	a1i 11	. . . . . . . . . . 11 |- ( m e. NN -> RR*s e. _V )
85		ssid 3624	. . . . . . . . . . . 12 |- RR* C_ RR*
86	85	a1i 11	. . . . . . . . . . 11 |- ( m e. NN -> RR* C_ RR* )
87		simp2 1062	. . . . . . . . . . . 12 |- ( ( m e. NN /\ x e. RR* /\ y e. RR* ) -> x e. RR* )
88		simp3 1063	. . . . . . . . . . . 12 |- ( ( m e. NN /\ x e. RR* /\ y e. RR* ) -> y e. RR* )
89	87, 88	xaddcld 12131	. . . . . . . . . . 11 |- ( ( m e. NN /\ x e. RR* /\ y e. RR* ) -> ( x +e y ) e. RR* )
90	16, 18, 26, 84, 86, 89	mulgnnsubcl 17553	. . . . . . . . . 10 |- ( ( m e. NN /\ m e. NN /\ B e. RR* ) -> ( m (.g ` RR*s ) B ) e. RR* )
91	90	3anidm12 1383	. . . . . . . . 9 |- ( ( m e. NN /\ B e. RR* ) -> ( m (.g ` RR*s ) B ) e. RR* )
92	91	ancoms 469	. . . . . . . 8 |- ( ( B e. RR* /\ m e. NN ) -> ( m (.g ` RR*s ) B ) e. RR* )
93		xrsinvgval 29677	. . . . . . . 8 |- ( ( m (.g ` RR*s ) B ) e. RR* -> ( ( invg ` RR*s ) ` ( m (.g ` RR*s ) B ) ) = -e ( m (.g ` RR*s ) B ) )
94	92, 93	syl 17	. . . . . . 7 |- ( ( B e. RR* /\ m e. NN ) -> ( ( invg ` RR*s ) ` ( m (.g ` RR*s ) B ) ) = -e ( m (.g ` RR*s ) B ) )
95	82, 94	eqtrd 2656	. . . . . 6 |- ( ( B e. RR* /\ m e. NN ) -> ( -u m (.g ` RR*s ) B ) = -e ( m (.g ` RR*s ) B ) )
96	95	adantr 481	. . . . 5 |- ( ( ( B e. RR* /\ m e. NN ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( -u m (.g ` RR*s ) B ) = -e ( m (.g ` RR*s ) B ) )
97		nnre 11027	. . . . . . . . . 10 |- ( m e. NN -> m e. RR )
98	97	adantl 482	. . . . . . . . 9 |- ( ( B e. RR* /\ m e. NN ) -> m e. RR )
99		rexneg 12042	. . . . . . . . 9 |- ( m e. RR -> -e m = -u m )
100	98, 99	syl 17	. . . . . . . 8 |- ( ( B e. RR* /\ m e. NN ) -> -e m = -u m )
101	100	oveq1d 6665	. . . . . . 7 |- ( ( B e. RR* /\ m e. NN ) -> ( -e m xe B ) = ( -u m xe B ) )
102		nnssre 11024	. . . . . . . . . 10 |- NN C_ RR
103	102, 53	sstri 3612	. . . . . . . . 9 |- NN C_ RR*
104		simpr 477	. . . . . . . . 9 |- ( ( B e. RR* /\ m e. NN ) -> m e. NN )
105	103, 104	sseldi 3601	. . . . . . . 8 |- ( ( B e. RR* /\ m e. NN ) -> m e. RR* )
106		simpl 473	. . . . . . . 8 |- ( ( B e. RR* /\ m e. NN ) -> B e. RR* )
107		xmulneg1 12099	. . . . . . . 8 |- ( ( m e. RR* /\ B e. RR* ) -> ( -e m xe B ) = -e ( m xe B ) )
108	105, 106, 107	syl2anc 693	. . . . . . 7 |- ( ( B e. RR* /\ m e. NN ) -> ( -e m xe B ) = -e ( m xe B ) )
109	101, 108	eqtr3d 2658	. . . . . 6 |- ( ( B e. RR* /\ m e. NN ) -> ( -u m xe B ) = -e ( m xe B ) )
110	109	adantr 481	. . . . 5 |- ( ( ( B e. RR* /\ m e. NN ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( -u m xe B ) = -e ( m xe B ) )
111	79, 96, 110	3eqtr4d 2666	. . . 4 |- ( ( ( B e. RR* /\ m e. NN ) /\ ( m (.g ` RR*s ) B ) = ( m xe B ) ) -> ( -u m (.g ` RR*s ) B ) = ( -u m xe B ) )
112	111	exp31 630	. . 3 |- ( B e. RR* -> ( m e. NN -> ( ( m (.g ` RR*s ) B ) = ( m xe B ) -> ( -u m (.g ` RR*s ) B ) = ( -u m xe B ) ) ) )
113	3, 6, 9, 12, 15, 21, 77, 112	zindd 11478	. 2 |- ( B e. RR* -> ( A e. ZZ -> ( A (.g ` RR*s ) B ) = ( A xe B ) ) )
114	113	impcom 446	1 |- ( ( A e. ZZ /\ B e. RR* ) -> ( A (.g ` RR*s ) B ) = ( A xe B ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   RR*cxr 10073   <_ cle 10075   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   -ecxne 11943   +ecxad 11944   xecxmu 11945   RR*scxrs 16160   invgcminusg 17423  ._gcmg 17540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-fz 12327  df-seq 12802  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-tset 15960  df-ple 15961  df-ds 15964  df-0g 16102  df-xrs 16162  df-minusg 17426  df-mulg 17541

This theorem is referenced by:  xrge0mulgnn0  29689  pnfinf  29737