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Unicode version
Theorem muladdOLD 6583
Description: Product of two sums.
Assertion
Ref Expression
muladdOLD |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
Proof of Theorem muladdOLD
StepHypRef Expression
1 addcl 6454 . . . 4 |- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
21adantr 425 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (A + B) e. CC)
3 simprl 450 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> C e. CC)
4 simprr 451 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> D e. CC)
5 adddi 6462 . . 3 |- (((A + B) e. CC /\ C e. CC /\ D e. CC) -> ((A + B) x. (C + D)) = (((A + B) x. C) + ((A + B) x. D)))
62, 3, 4, 5syl111anc 1100 . 2 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. (C + D)) = (((A + B) x. C) + ((A + B) x. D)))
7 adddir 6472 . . . . 5 |- ((A e. CC /\ B e. CC /\ C e. CC) -> ((A + B) x. C) = ((A x. C) + (B x. C)))
873expa 1067 . . . 4 |- (((A e. CC /\ B e. CC) /\ C e. CC) -> ((A + B) x. C) = ((A x. C) + (B x. C)))
98adantrr 431 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. C) = ((A x. C) + (B x. C)))
10 adddir 6472 . . . . 5 |- ((A e. CC /\ B e. CC /\ D e. CC) -> ((A + B) x. D) = ((A x. D) + (B x. D)))
11103expa 1067 . . . 4 |- (((A e. CC /\ B e. CC) /\ D e. CC) -> ((A + B) x. D) = ((A x. D) + (B x. D)))
1211adantrl 430 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. D) = ((A x. D) + (B x. D)))
139, 12opreq12d 4900 . 2 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A + B) x. C) + ((A + B) x. D)) = (((A x. C) + (B x. C)) + ((A x. D) + (B x. D))))
14 mulcl 6456 . . . . 5 |- ((A e. CC /\ C e. CC) -> (A x. C) e. CC)
1514ad2ant2r 445 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (A x. C) e. CC)
16 mulcl 6456 . . . . 5 |- ((B e. CC /\ C e. CC) -> (B x. C) e. CC)
1716ad2ant2lr 446 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (B x. C) e. CC)
18 addcl 6454 . . . . . . 7 |- (((A x. D) e. CC /\ (B x. D) e. CC) -> ((A x. D) + (B x. D)) e. CC)
19 mulcl 6456 . . . . . . 7 |- ((A e. CC /\ D e. CC) -> (A x. D) e. CC)
20 mulcl 6456 . . . . . . 7 |- ((B e. CC /\ D e. CC) -> (B x. D) e. CC)
2118, 19, 20syl2an 503 . . . . . 6 |- (((A e. CC /\ D e. CC) /\ (B e. CC /\ D e. CC)) -> ((A x. D) + (B x. D)) e. CC)
2221anandirs 571 . . . . 5 |- (((A e. CC /\ B e. CC) /\ D e. CC) -> ((A x. D) + (B x. D)) e. CC)
2322adantrl 430 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. D) + (B x. D)) e. CC)
24 add23 6490 . . . 4 |- (((A x. C) e. CC /\ (B x. C) e. CC /\ ((A x. D) + (B x. D)) e. CC) -> (((A x. C) + (B x. C)) + ((A x. D) + (B x. D))) = (((A x. C) + ((A x. D) + (B x. D))) + (B x. C)))
2515, 17, 23, 24syl111anc 1100 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (B x. C)) + ((A x. D) + (B x. D))) = (((A x. C) + ((A x. D) + (B x. D))) + (B x. C)))
26 mulcom 6459 . . . . . . 7 |- ((B e. CC /\ D e. CC) -> (B x. D) = (D x. B))
2726ad2ant2l 444 . . . . . 6 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (B x. D) = (D x. B))
2827opreq2d 4898 . . . . 5 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (A x. D)) + (B x. D)) = (((A x. C) + (A x. D)) + (D x. B)))
2919ad2ant2rl 447 . . . . . 6 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (A x. D) e. CC)
3020ad2ant2l 444 . . . . . 6 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (B x. D) e. CC)
31 addass 6460 . . . . . 6 |- (((A x. C) e. CC /\ (A x. D) e. CC /\ (B x. D) e. CC) -> (((A x. C) + (A x. D)) + (B x. D)) = ((A x. C) + ((A x. D) + (B x. D))))
3215, 29, 30, 31syl111anc 1100 . . . . 5 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (A x. D)) + (B x. D)) = ((A x. C) + ((A x. D) + (B x. D))))
33 mulcl 6456 . . . . . . . 8 |- ((D e. CC /\ B e. CC) -> (D x. B) e. CC)
3433ancoms 484 . . . . . . 7 |- ((B e. CC /\ D e. CC) -> (D x. B) e. CC)
3534ad2ant2l 444 . . . . . 6 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (D x. B) e. CC)
36 add23 6490 . . . . . 6 |- (((A x. C) e. CC /\ (A x. D) e. CC /\ (D x. B) e. CC) -> (((A x. C) + (A x. D)) + (D x. B)) = (((A x. C) + (D x. B)) + (A x. D)))
3715, 29, 35, 36syl111anc 1100 . . . . 5 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (A x. D)) + (D x. B)) = (((A x. C) + (D x. B)) + (A x. D)))
3828, 32, 373eqtr3d 1934 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. C) + ((A x. D) + (B x. D))) = (((A x. C) + (D x. B)) + (A x. D)))
39 mulcom 6459 . . . . 5 |- ((B e. CC /\ C e. CC) -> (B x. C) = (C x. B))
4039ad2ant2lr 446 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (B x. C) = (C x. B))
4138, 40opreq12d 4900 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + ((A x. D) + (B x. D))) + (B x. C)) = ((((A x. C) + (D x. B)) + (A x. D)) + (C x. B)))
42 addcl 6454 . . . . . 6 |- (((A x. C) e. CC /\ (D x. B) e. CC) -> ((A x. C) + (D x. B)) e. CC)
4342, 14, 34syl2an 503 . . . . 5 |- (((A e. CC /\ C e. CC) /\ (B e. CC /\ D e. CC)) -> ((A x. C) + (D x. B)) e. CC)
4443an4s 566 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A x. C) + (D x. B)) e. CC)
45 mulcl 6456 . . . . . 6 |- ((C e. CC /\ B e. CC) -> (C x. B) e. CC)
4645ancoms 484 . . . . 5 |- ((B e. CC /\ C e. CC) -> (C x. B) e. CC)
4746ad2ant2lr 446 . . . 4 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (C x. B) e. CC)
48 addass 6460 . . . 4 |- ((((A x. C) + (D x. B)) e. CC /\ (A x. D) e. CC /\ (C x. B) e. CC) -> ((((A x. C) + (D x. B)) + (A x. D)) + (C x. B)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
4944, 29, 47, 48syl111anc 1100 . . 3 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((((A x. C) + (D x. B)) + (A x. D)) + (C x. B)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
5025, 41, 493eqtrd 1929 . 2 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> (((A x. C) + (B x. C)) + ((A x. D) + (B x. D))) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
516, 13, 503eqtrd 1929 1 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. CC)) -> ((A + B) x. (C + D)) = (((A x. C) + (D x. B)) + ((A x. D) + (C x. B))))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  (class class class)co 4884  CCcc 6384   + caddc 6389   x. cmul 6391
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-m1r 6325  df-c 6392  df-plus 6397  df-mul 6398
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