Theorem xadd4d 10119

Description: Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8347. (Contributed by Alexander van der Vekens, 21-Dec-2017.)

Hypotheses

Ref	Expression
xadd4d.1	|- ( ph -> ( A e. RR* /\ A =/= -oo ) )
xadd4d.2	|- ( ph -> ( B e. RR* /\ B =/= -oo ) )
xadd4d.3	|- ( ph -> ( C e. RR* /\ C =/= -oo ) )
xadd4d.4	|- ( ph -> ( D e. RR* /\ D =/= -oo ) )

Assertion

Ref	Expression
xadd4d	|- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )

Proof of Theorem xadd4d

Step	Hyp	Ref	Expression
1		xadd4d.3	. . . 4 |- ( ph -> ( C e. RR* /\ C =/= -oo ) )
2		xadd4d.2	. . . 4 |- ( ph -> ( B e. RR* /\ B =/= -oo ) )
3		xadd4d.4	. . . 4 |- ( ph -> ( D e. RR* /\ D =/= -oo ) )
4		xaddass 10103	. . . 4 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( ( C +e B ) +e D ) = ( C +e ( B +e D ) ) )
5	1, 2, 3, 4	syl3anc 1273	. . 3 |- ( ph -> ( ( C +e B ) +e D ) = ( C +e ( B +e D ) ) )
6	5	oveq2d 6033	. 2 |- ( ph -> ( A +e ( ( C +e B ) +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
7		xadd4d.1	. . . 4 |- ( ph -> ( A e. RR* /\ A =/= -oo ) )
8	1	simpld 112	. . . . 5 |- ( ph -> C e. RR* )
9	3	simpld 112	. . . . 5 |- ( ph -> D e. RR* )
10	8, 9	xaddcld 10118	. . . 4 |- ( ph -> ( C +e D ) e. RR* )
11		xaddnemnf 10091	. . . . 5 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( C +e D ) =/= -oo )
12	1, 3, 11	syl2anc 411	. . . 4 |- ( ph -> ( C +e D ) =/= -oo )
13		xaddass 10103	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( B e. RR* /\ B =/= -oo ) /\ ( ( C +e D ) e. RR* /\ ( C +e D ) =/= -oo ) ) -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( B +e ( C +e D ) ) ) )
14	7, 2, 10, 12, 13	syl112anc 1277	. . 3 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( B +e ( C +e D ) ) ) )
15	2	simpld 112	. . . . . . 7 |- ( ph -> B e. RR* )
16		xaddcom 10095	. . . . . . 7 |- ( ( C e. RR* /\ B e. RR* ) -> ( C +e B ) = ( B +e C ) )
17	8, 15, 16	syl2anc 411	. . . . . 6 |- ( ph -> ( C +e B ) = ( B +e C ) )
18	17	oveq1d 6032	. . . . 5 |- ( ph -> ( ( C +e B ) +e D ) = ( ( B +e C ) +e D ) )
19		xaddass 10103	. . . . . 6 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( ( B +e C ) +e D ) = ( B +e ( C +e D ) ) )
20	2, 1, 3, 19	syl3anc 1273	. . . . 5 |- ( ph -> ( ( B +e C ) +e D ) = ( B +e ( C +e D ) ) )
21	18, 20	eqtr2d 2265	. . . 4 |- ( ph -> ( B +e ( C +e D ) ) = ( ( C +e B ) +e D ) )
22	21	oveq2d 6033	. . 3 |- ( ph -> ( A +e ( B +e ( C +e D ) ) ) = ( A +e ( ( C +e B ) +e D ) ) )
23	14, 22	eqtrd 2264	. 2 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( ( C +e B ) +e D ) ) )
24	15, 9	xaddcld 10118	. . 3 |- ( ph -> ( B +e D ) e. RR* )
25		xaddnemnf 10091	. . . 4 |- ( ( ( B e. RR* /\ B =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( B +e D ) =/= -oo )
26	2, 3, 25	syl2anc 411	. . 3 |- ( ph -> ( B +e D ) =/= -oo )
27		xaddass 10103	. . 3 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ ( C e. RR* /\ C =/= -oo ) /\ ( ( B +e D ) e. RR* /\ ( B +e D ) =/= -oo ) ) -> ( ( A +e C ) +e ( B +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
28	7, 1, 24, 26, 27	syl112anc 1277	. 2 |- ( ph -> ( ( A +e C ) +e ( B +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
29	6, 23, 28	3eqtr4d 2274	1 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   =/= wne 2402  (class class class)co 6017   -oocmnf 8211   RR*cxr 8212   +ecxad 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128  ax-addcom 8131  ax-addass 8133  ax-rnegex 8140

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-xadd 10007

This theorem is referenced by:  xnn0add4d  10120