Theorem omltoe 43852

Description: Exponentiation eventually dominates multiplication. (Contributed by RP, 3-Jan-2025.)

Assertion

Ref	Expression
omltoe	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴)))

Proof of Theorem omltoe

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
2	1	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 𝐵 ∈ On)
3		oe2 43851	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 ·o 𝐵) = (𝐵 ↑o 2o))
4	2, 3	syl 17	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐵) = (𝐵 ↑o 2o))
5		2on 8411	. . . . . . . . 9 ⊢ 2o ∈ On
6	5	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o ∈ On)
7		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
8	6, 7, 1	3jca 1129	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
9	8	adantr 480	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
10		simpr 484	. . . . . . . . 9 ⊢ ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
11	10	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 𝐴 ∈ 𝐵)
12	11	ne0d 4283	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 𝐵 ≠ ∅)
13		on0eln0 6374	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
14	2, 13	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (∅ ∈ 𝐵 ↔ 𝐵 ≠ ∅))
15	12, 14	mpbird 257	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → ∅ ∈ 𝐵)
16	9, 15	jca 511	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → ((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵))
17		df-2o 8399	. . . . . . 7 ⊢ 2o = suc 1o
18	17	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc 1o)
19		simpl 482	. . . . . . . . 9 ⊢ ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 1o ∈ 𝐴)
20	19	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴)
21		eloni 6327	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
23	22	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴)
24	20, 23	jca 511	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴))
25		ordelsuc 7764	. . . . . . . 8 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o ⊆ 𝐴))
26	25	biimpd 229	. . . . . . 7 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 → suc 1o ⊆ 𝐴))
27	24, 20, 26	sylc 65	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → suc 1o ⊆ 𝐴)
28	18, 27	eqsstrd 3957	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴)
29		oewordi 8520	. . . . 5 ⊢ (((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (2o ⊆ 𝐴 → (𝐵 ↑o 2o) ⊆ (𝐵 ↑o 𝐴)))
30	16, 28, 29	sylc 65	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ↑o 2o) ⊆ (𝐵 ↑o 𝐴))
31	4, 30	eqsstrd 3957	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐵) ⊆ (𝐵 ↑o 𝐴))
32	2, 2, 15	jca31 514	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → ((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵))
33		omordi 8494	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (𝐴 ∈ 𝐵 → (𝐵 ·o 𝐴) ∈ (𝐵 ·o 𝐵)))
34	32, 11, 33	sylc 65	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐴) ∈ (𝐵 ·o 𝐵))
35	31, 34	sseldd 3923	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴))
36	35	ex 412	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 ·o 𝐴) ∈ (𝐵 ↑o 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ⊆ wss 3890  ∅c0 4274  Ord word 6316  Oncon0 6317  suc csuc 6319  (class class class)co 7360  1_oc1o 8391  2_oc2o 8392   ·_o comu 8396   ↑_o coe 8397

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-omul 8403  df-oexp 8404

This theorem is referenced by: (None)