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Theorem omlimcl2 41552
Description: The product of a limit ordinal with any nonzero ordinal is a limit ordinal. (Contributed by RP, 8-Jan-2025.)
Assertion
Ref Expression
omlimcl2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))

Proof of Theorem omlimcl2
StepHypRef Expression
1 eloni 6326 . . . . . 6 (𝐴 ∈ On → Ord 𝐴)
21ad2antrr 724 . . . . 5 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Ord 𝐴)
3 ne0i 4293 . . . . . 6 (∅ ∈ 𝐴𝐴 ≠ ∅)
43adantl 482 . . . . 5 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ≠ ∅)
5 id 22 . . . . 5 (𝐴 = 𝐴𝐴 = 𝐴)
6 df-lim 6321 . . . . . 6 (Lim 𝐴 ↔ (Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴))
76biimpri 227 . . . . 5 ((Ord 𝐴𝐴 ≠ ∅ ∧ 𝐴 = 𝐴) → Lim 𝐴)
82, 4, 5, 7syl2an3an 1422 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = 𝐴) → Lim 𝐴)
98ex 413 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = 𝐴 → Lim 𝐴))
10 limelon 6380 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → 𝐵 ∈ On)
1110ad3antlr 729 . . . . 5 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → 𝐵 ∈ On)
12 simpll 765 . . . . . 6 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → 𝐴 ∈ On)
1312anim1i 615 . . . . 5 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → (𝐴 ∈ On ∧ Lim 𝐴))
14 0ellim 6379 . . . . . . 7 (Lim 𝐵 → ∅ ∈ 𝐵)
1514adantl 482 . . . . . 6 ((𝐵𝐶 ∧ Lim 𝐵) → ∅ ∈ 𝐵)
1615ad3antlr 729 . . . . 5 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → ∅ ∈ 𝐵)
17 omlimcl 8522 . . . . 5 (((𝐵 ∈ On ∧ (𝐴 ∈ On ∧ Lim 𝐴)) ∧ ∅ ∈ 𝐵) → Lim (𝐵 ·o 𝐴))
1811, 13, 16, 17syl21anc 836 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ Lim 𝐴) → Lim (𝐵 ·o 𝐴))
1918ex 413 . . 3 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (Lim 𝐴 → Lim (𝐵 ·o 𝐴)))
209, 19syld 47 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = 𝐴 → Lim (𝐵 ·o 𝐴)))
21 onuni 7720 . . . . . . . . 9 (𝐴 ∈ On → 𝐴 ∈ On)
2221, 10anim12ci 614 . . . . . . . 8 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
23 omcl 8479 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ·o 𝐴) ∈ On)
2422, 23syl 17 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐵 ·o 𝐴) ∈ On)
25 simpr 485 . . . . . . 7 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → (𝐵𝐶 ∧ Lim 𝐵))
2624, 25jca 512 . . . . . 6 ((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → ((𝐵 ·o 𝐴) ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)))
2726ad2antrr 724 . . . . 5 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → ((𝐵 ·o 𝐴) ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)))
28 oalimcl 8504 . . . . 5 (((𝐵 ·o 𝐴) ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) → Lim ((𝐵 ·o 𝐴) +o 𝐵))
2927, 28syl 17 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → Lim ((𝐵 ·o 𝐴) +o 𝐵))
30 simpr 485 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → 𝐴 = suc 𝐴)
3130oveq2d 7370 . . . . . 6 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → (𝐵 ·o 𝐴) = (𝐵 ·o suc 𝐴))
3222ad2antrr 724 . . . . . . 7 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → (𝐵 ∈ On ∧ 𝐴 ∈ On))
33 omsuc 8469 . . . . . . 7 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ·o suc 𝐴) = ((𝐵 ·o 𝐴) +o 𝐵))
3432, 33syl 17 . . . . . 6 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → (𝐵 ·o suc 𝐴) = ((𝐵 ·o 𝐴) +o 𝐵))
3531, 34eqtrd 2776 . . . . 5 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → (𝐵 ·o 𝐴) = ((𝐵 ·o 𝐴) +o 𝐵))
36 limeq 6328 . . . . 5 ((𝐵 ·o 𝐴) = ((𝐵 ·o 𝐴) +o 𝐵) → (Lim (𝐵 ·o 𝐴) ↔ Lim ((𝐵 ·o 𝐴) +o 𝐵)))
3735, 36syl 17 . . . 4 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → (Lim (𝐵 ·o 𝐴) ↔ Lim ((𝐵 ·o 𝐴) +o 𝐵)))
3829, 37mpbird 256 . . 3 ((((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) ∧ 𝐴 = suc 𝐴) → Lim (𝐵 ·o 𝐴))
3938ex 413 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = suc 𝐴 → Lim (𝐵 ·o 𝐴)))
40 orduniorsuc 7762 . . 3 (Ord 𝐴 → (𝐴 = 𝐴𝐴 = suc 𝐴))
412, 40syl 17 . 2 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → (𝐴 = 𝐴𝐴 = suc 𝐴))
4220, 39, 41mpjaod 858 1 (((𝐴 ∈ On ∧ (𝐵𝐶 ∧ Lim 𝐵)) ∧ ∅ ∈ 𝐴) → Lim (𝐵 ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2942  c0 4281   cuni 4864  Ord word 6315  Oncon0 6316  Lim wlim 6317  suc csuc 6318  (class class class)co 7354   +o coa 8406   ·o comu 8407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6252  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7800  df-2nd 7919  df-frecs 8209  df-wrecs 8240  df-recs 8314  df-rdg 8353  df-1o 8409  df-oadd 8413  df-omul 8414
This theorem is referenced by:  onexlimgt  41553  succlg  41638  dflim5  41639
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