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Theorem omsuc 6368
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 4293 . . . . . . 7 suc 𝐵 = (𝐵 ∪ {𝐵})
2 iuneq1 3826 . . . . . . 7 (suc 𝐵 = (𝐵 ∪ {𝐵}) → 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·o 𝑥) +o 𝐴))
31, 2ax-mp 5 . . . . . 6 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·o 𝑥) +o 𝐴)
4 iunxun 3892 . . . . . 6 𝑥 ∈ (𝐵 ∪ {𝐵})((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴))
53, 4eqtri 2160 . . . . 5 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴))
6 oveq2 5782 . . . . . . . 8 (𝑥 = 𝐵 → (𝐴 ·o 𝑥) = (𝐴 ·o 𝐵))
76oveq1d 5789 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
87iunxsng 3888 . . . . . 6 (𝐵 ∈ On → 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴) = ((𝐴 ·o 𝐵) +o 𝐴))
98uneq2d 3230 . . . . 5 (𝐵 ∈ On → ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ 𝑥 ∈ {𝐵} ((𝐴 ·o 𝑥) +o 𝐴)) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
105, 9syl5eq 2184 . . . 4 (𝐵 ∈ On → 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
1110adantl 275 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
12 suceloni 4417 . . . 4 (𝐵 ∈ On → suc 𝐵 ∈ On)
13 omv2 6361 . . . 4 ((𝐴 ∈ On ∧ suc 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴))
1412, 13sylan2 284 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = 𝑥 ∈ suc 𝐵((𝐴 ·o 𝑥) +o 𝐴))
15 omv2 6361 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) = 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴))
1615uneq1d 3229 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ( 𝑥𝐵 ((𝐴 ·o 𝑥) +o 𝐴) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
1711, 14, 163eqtr4d 2182 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)))
18 omcl 6357 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
19 simpl 108 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
20 oaword1 6367 . . . 4 (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → (𝐴 ·o 𝐵) ⊆ ((𝐴 ·o 𝐵) +o 𝐴))
21 ssequn1 3246 . . . 4 ((𝐴 ·o 𝐵) ⊆ ((𝐴 ·o 𝐵) +o 𝐴) ↔ ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
2220, 21sylib 121 . . 3 (((𝐴 ·o 𝐵) ∈ On ∧ 𝐴 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
2318, 19, 22syl2anc 408 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 𝐵) ∪ ((𝐴 ·o 𝐵) +o 𝐴)) = ((𝐴 ·o 𝐵) +o 𝐴))
2417, 23eqtrd 2172 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 103   = wceq 1331  wcel 1480  cun 3069  wss 3071  {csn 3527   ciun 3813  Oncon0 4285  suc csuc 4287  (class class class)co 5774   +o coa 6310   ·o comu 6311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-oadd 6317  df-omul 6318
This theorem is referenced by:  onmsuc  6369
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