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Theorem omltoe 42760
Description: Exponentiation eventually dominates multiplication. (Contributed by RP, 3-Jan-2025.)
Assertion
Ref Expression
omltoe ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o𝐴𝐴𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵o 𝐴)))

Proof of Theorem omltoe
StepHypRef Expression
1 simpr 484 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
21adantr 480 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 𝐵 ∈ On)
3 oe2 42759 . . . . 5 (𝐵 ∈ On → (𝐵 ·o 𝐵) = (𝐵o 2o))
42, 3syl 17 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 ·o 𝐵) = (𝐵o 2o))
5 2on 8494 . . . . . . . . 9 2o ∈ On
65a1i 11 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o ∈ On)
7 simpl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
86, 7, 13jca 1126 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
98adantr 480 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
10 simpr 484 . . . . . . . . 9 ((1o𝐴𝐴𝐵) → 𝐴𝐵)
1110adantl 481 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 𝐴𝐵)
1211ne0d 4331 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 𝐵 ≠ ∅)
13 on0eln0 6419 . . . . . . . 8 (𝐵 ∈ On → (∅ ∈ 𝐵𝐵 ≠ ∅))
142, 13syl 17 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (∅ ∈ 𝐵𝐵 ≠ ∅))
1512, 14mpbird 257 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → ∅ ∈ 𝐵)
169, 15jca 511 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → ((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵))
17 df-2o 8481 . . . . . . 7 2o = suc 1o
1817a1i 11 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 2o = suc 1o)
19 simpl 482 . . . . . . . . 9 ((1o𝐴𝐴𝐵) → 1o𝐴)
2019adantl 481 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 1o𝐴)
21 eloni 6373 . . . . . . . . . 10 (𝐴 ∈ On → Ord 𝐴)
2221adantr 480 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
2322adantr 480 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → Ord 𝐴)
2420, 23jca 511 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (1o𝐴 ∧ Ord 𝐴))
25 ordelsuc 7817 . . . . . . . 8 ((1o𝐴 ∧ Ord 𝐴) → (1o𝐴 ↔ suc 1o𝐴))
2625biimpd 228 . . . . . . 7 ((1o𝐴 ∧ Ord 𝐴) → (1o𝐴 → suc 1o𝐴))
2724, 20, 26sylc 65 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → suc 1o𝐴)
2818, 27eqsstrd 4016 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → 2o𝐴)
29 oewordi 8605 . . . . 5 (((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (2o𝐴 → (𝐵o 2o) ⊆ (𝐵o 𝐴)))
3016, 28, 29sylc 65 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵o 2o) ⊆ (𝐵o 𝐴))
314, 30eqsstrd 4016 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 ·o 𝐵) ⊆ (𝐵o 𝐴))
322, 2, 15jca31 514 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → ((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵))
33 omordi 8580 . . . 4 (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴𝐵 → (𝐵 ·o 𝐴) ∈ (𝐵 ·o 𝐵)))
3432, 11, 33sylc 65 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 ·o 𝐴) ∈ (𝐵 ·o 𝐵))
3531, 34sseldd 3979 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o𝐴𝐴𝐵)) → (𝐵 ·o 𝐴) ∈ (𝐵o 𝐴))
3635ex 412 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o𝐴𝐴𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵o 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wss 3944  c0 4318  Ord word 6362  Oncon0 6363  suc csuc 6365  (class class class)co 7414  1oc1o 8473  2oc2o 8474   ·o comu 8478  o coe 8479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-oexp 8486
This theorem is referenced by: (None)
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