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Theorem omsuc 5990
Description: Multiplication with successor. Definition 8.15 of [TakeutiZaring] p. 62. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
omsuc ((A On B On) → (A ·𝑜 suc B) = ((A ·𝑜 B) +𝑜 A))

Proof of Theorem omsuc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 df-suc 4074 . . . . . . 7 suc B = (B ∪ {B})
2 iuneq1 3661 . . . . . . 7 (suc B = (B ∪ {B}) → x suc B((A ·𝑜 x) +𝑜 A) = x (B ∪ {B})((A ·𝑜 x) +𝑜 A))
31, 2ax-mp 7 . . . . . 6 x suc B((A ·𝑜 x) +𝑜 A) = x (B ∪ {B})((A ·𝑜 x) +𝑜 A)
4 iunxun 3726 . . . . . 6 x (B ∪ {B})((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ x {B} ((A ·𝑜 x) +𝑜 A))
53, 4eqtri 2057 . . . . 5 x suc B((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ x {B} ((A ·𝑜 x) +𝑜 A))
6 oveq2 5463 . . . . . . . 8 (x = B → (A ·𝑜 x) = (A ·𝑜 B))
76oveq1d 5470 . . . . . . 7 (x = B → ((A ·𝑜 x) +𝑜 A) = ((A ·𝑜 B) +𝑜 A))
87iunxsng 3723 . . . . . 6 (B On → x {B} ((A ·𝑜 x) +𝑜 A) = ((A ·𝑜 B) +𝑜 A))
98uneq2d 3091 . . . . 5 (B On → ( x B ((A ·𝑜 x) +𝑜 A) ∪ x {B} ((A ·𝑜 x) +𝑜 A)) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
105, 9syl5eq 2081 . . . 4 (B On → x suc B((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
1110adantl 262 . . 3 ((A On B On) → x suc B((A ·𝑜 x) +𝑜 A) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
12 suceloni 4193 . . . 4 (B On → suc B On)
13 omv2 5984 . . . 4 ((A On suc B On) → (A ·𝑜 suc B) = x suc B((A ·𝑜 x) +𝑜 A))
1412, 13sylan2 270 . . 3 ((A On B On) → (A ·𝑜 suc B) = x suc B((A ·𝑜 x) +𝑜 A))
15 omv2 5984 . . . 4 ((A On B On) → (A ·𝑜 B) = x B ((A ·𝑜 x) +𝑜 A))
1615uneq1d 3090 . . 3 ((A On B On) → ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ( x B ((A ·𝑜 x) +𝑜 A) ∪ ((A ·𝑜 B) +𝑜 A)))
1711, 14, 163eqtr4d 2079 . 2 ((A On B On) → (A ·𝑜 suc B) = ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)))
18 omcl 5980 . . 3 ((A On B On) → (A ·𝑜 B) On)
19 simpl 102 . . 3 ((A On B On) → A On)
20 oaword1 5989 . . . 4 (((A ·𝑜 B) On A On) → (A ·𝑜 B) ⊆ ((A ·𝑜 B) +𝑜 A))
21 ssequn1 3107 . . . 4 ((A ·𝑜 B) ⊆ ((A ·𝑜 B) +𝑜 A) ↔ ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ((A ·𝑜 B) +𝑜 A))
2220, 21sylib 127 . . 3 (((A ·𝑜 B) On A On) → ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ((A ·𝑜 B) +𝑜 A))
2318, 19, 22syl2anc 391 . 2 ((A On B On) → ((A ·𝑜 B) ∪ ((A ·𝑜 B) +𝑜 A)) = ((A ·𝑜 B) +𝑜 A))
2417, 23eqtrd 2069 1 ((A On B On) → (A ·𝑜 suc B) = ((A ·𝑜 B) +𝑜 A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cun 2909  wss 2911  {csn 3367   ciun 3648  Oncon0 4066  suc csuc 4068  (class class class)co 5455   +𝑜 coa 5937   ·𝑜 comu 5938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945
This theorem is referenced by:  onmsuc  5991
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