Theorem oaltom 44058

Description: Multiplication eventually dominates addition. (Contributed by RP, 3-Jan-2025.)

Assertion

Ref	Expression
oaltom	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))

Proof of Theorem oaltom

Step	Hyp	Ref	Expression
1		om2 8571	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
2	1	ad2antlr 739	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
3		2on 8467	. . . . . . . 8 ⊢ 2o ∈ On
4	3	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o ∈ On)
5		simpl 487	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6		simpr 489	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
7	4, 5, 6	3jca 1144	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
8	7	adantr 485	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
9		df-2o 8454	. . . . . . 7 ⊢ 2o = suc 1o
10	9	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc 1o)
11		simprl 782	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴)
12		eloni 6371	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
13	12	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
14	13	adantr 485	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴)
15	11, 14	jca 520	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴))
16		ordelsuc 7816	. . . . . . . 8 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o ⊆ 𝐴))
17	16	biimpd 232	. . . . . . 7 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 → suc 1o ⊆ 𝐴))
18	15, 11, 17	sylc 66	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → suc 1o ⊆ 𝐴)
19	10, 18	eqsstrd 3977	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴)
20		omwordi 8556	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 → (𝐵 ·o 2o) ⊆ (𝐵 ·o 𝐴)))
21	8, 19, 20	sylc 66	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 2o) ⊆ (𝐵 ·o 𝐴))
22	2, 21	eqsstrd 3977	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) ⊆ (𝐵 ·o 𝐴))
23	6, 6	jca 520	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐵 ∈ On))
24		simpr 489	. . . 4 ⊢ ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
25		oaordi 8531	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)))
26	25	imp 411	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
27	23, 24, 26	syl2an 607	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
28	22, 27	sseldd 3944	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴))
29	28	ex 417	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3911  Ord word 6360  Oncon0 6361  suc csuc 6363  (class class class)co 7411  1_oc1o 8446  2_oc2o 8447   +_o coa 8450   ·_o comu 8451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-omul 8458

This theorem is referenced by: (None)