Theorem xadd4d 9954

Description: Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8190. (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 9938	. . . 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 1249	. . 3 |- ( ph -> ( ( C +e B ) +e D ) = ( C +e ( B +e D ) ) )
6	5	oveq2d 5935	. 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 9953	. . . 4 |- ( ph -> ( C +e D ) e. RR* )
11		xaddnemnf 9926	. . . . 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 9938	. . . 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 1253	. . 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 9930	. . . . . . 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 5934	. . . . 5 |- ( ph -> ( ( C +e B ) +e D ) = ( ( B +e C ) +e D ) )
19		xaddass 9938	. . . . . 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 1249	. . . . 5 |- ( ph -> ( ( B +e C ) +e D ) = ( B +e ( C +e D ) ) )
21	18, 20	eqtr2d 2227	. . . 4 |- ( ph -> ( B +e ( C +e D ) ) = ( ( C +e B ) +e D ) )
22	21	oveq2d 5935	. . 3 |- ( ph -> ( A +e ( B +e ( C +e D ) ) ) = ( A +e ( ( C +e B ) +e D ) ) )
23	14, 22	eqtrd 2226	. 2 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( ( C +e B ) +e D ) ) )
24	15, 9	xaddcld 9953	. . 3 |- ( ph -> ( B +e D ) e. RR* )
25		xaddnemnf 9926	. . . 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 9938	. . 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 1253	. 2 |- ( ph -> ( ( A +e C ) +e ( B +e D ) ) = ( A +e ( C +e ( B +e D ) ) ) )
29	6, 23, 28	3eqtr4d 2236	1 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( ( A +e C ) +e ( B +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   =/= wne 2364  (class class class)co 5919   -oocmnf 8054   RR*cxr 8055   +ecxad 9839

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971  ax-addcom 7974  ax-addass 7976  ax-rnegex 7983

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-xadd 9842

This theorem is referenced by:  xnn0add4d  9955