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Theorem oaltom 43418
Description: Multiplication eventually dominates addition. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
oaltom ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o𝐴𝐴𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))

Proof of Theorem oaltom
StepHypRef Expression
1 om2 43417 . . . . 5 (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
21ad2antlr 727 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
3 2on 8520 . . . . . . . 8 2o ∈ On
43a1i 11 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o ∈ On)
5 simpl 482 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6 simpr 484 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
74, 5, 63jca 1129 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
87adantr 480 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
9 df-2o 8507 . . . . . . 7 2o = suc 1o
109a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 2o = suc 1o)
11 simprl 771 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 1o𝐴)
12 eloni 6394 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
1312adantr 480 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
1413adantr 480 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → Ord 𝐴)
1511, 14jca 511 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (1o𝐴 ∧ Ord 𝐴))
16 ordelsuc 7840 . . . . . . . 8 ((1o𝐴 ∧ Ord 𝐴) → (1o𝐴 ↔ suc 1o𝐴))
1716biimpd 229 . . . . . . 7 ((1o𝐴 ∧ Ord 𝐴) → (1o𝐴 → suc 1o𝐴))
1815, 11, 17sylc 65 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → suc 1o𝐴)
1910, 18eqsstrd 4018 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 2o𝐴)
20 omwordi 8609 . . . . 5 ((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o𝐴 → (𝐵 ·o 2o) ⊆ (𝐵 ·o 𝐴)))
218, 19, 20sylc 65 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 ·o 2o) ⊆ (𝐵 ·o 𝐴))
222, 21eqsstrd 4018 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 +o 𝐵) ⊆ (𝐵 ·o 𝐴))
236, 6jca 511 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐵 ∈ On))
24 simpr 484 . . . 4 ((1o𝐴𝐴𝐵) → 𝐴𝐵)
25 oaordi 8584 . . . . 5 ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)))
2625imp 406 . . . 4 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
2723, 24, 26syl2an 596 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
2822, 27sseldd 3984 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴))
2928ex 412 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o𝐴𝐴𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  Ord word 6383  Oncon0 6384  suc csuc 6386  (class class class)co 7431  1oc1o 8499  2oc2o 8500   +o coa 8503   ·o comu 8504
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511
This theorem is referenced by: (None)
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