Theorem xadd4d 9977

Description: Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 8212. (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 9961	. . . 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 5941	. 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 9976	. . . 4 |- ( ph -> ( C +e D ) e. RR* )
11		xaddnemnf 9949	. . . . 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 9961	. . . 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 9953	. . . . . . 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 5940	. . . . 5 |- ( ph -> ( ( C +e B ) +e D ) = ( ( B +e C ) +e D ) )
19		xaddass 9961	. . . . . 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 2230	. . . 4 |- ( ph -> ( B +e ( C +e D ) ) = ( ( C +e B ) +e D ) )
22	21	oveq2d 5941	. . 3 |- ( ph -> ( A +e ( B +e ( C +e D ) ) ) = ( A +e ( ( C +e B ) +e D ) ) )
23	14, 22	eqtrd 2229	. 2 |- ( ph -> ( ( A +e B ) +e ( C +e D ) ) = ( A +e ( ( C +e B ) +e D ) ) )
24	15, 9	xaddcld 9976	. . 3 |- ( ph -> ( B +e D ) e. RR* )
25		xaddnemnf 9949	. . . 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 9961	. . 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 2239	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 2167   =/= wne 2367  (class class class)co 5925   -oocmnf 8076   RR*cxr 8077   +ecxad 9862

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993  ax-addcom 7996  ax-addass 7998  ax-rnegex 8005

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-xadd 9865

This theorem is referenced by:  xnn0add4d  9978