Theorem xadd4d 9951

Description: Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8188. (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 9935	. . . 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 5934	. 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 9950	. . . 4 |- ( ph -> ( C +e D ) e. RR* )
11		xaddnemnf 9923	. . . . 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 9935	. . . 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 9927	. . . . . . 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 5933	. . . . 5 |- ( ph -> ( ( C +e B ) +e D ) = ( ( B +e C ) +e D ) )
19		xaddass 9935	. . . . . 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 5934	. . 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 9950	. . 3 |- ( ph -> ( B +e D ) e. RR* )
25		xaddnemnf 9923	. . . 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 9935	. . 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 5918   -oocmnf 8052   RR*cxr 8053   +ecxad 9836

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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969  ax-addcom 7972  ax-addass 7974  ax-rnegex 7981

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-xadd 9839

This theorem is referenced by:  xnn0add4d  9952