Theorem oaltom 43395

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 43394	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
2	1	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
3		2on 8519	. . . . . . . 8 ⊢ 2o ∈ On
4	3	a1i 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)
7	4, 5, 6	3jca 1127	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
8	7	adantr 480	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
9		df-2o 8506	. . . . . . 7 ⊢ 2o = suc 1o
10	9	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc 1o)
11		simprl 771	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴)
12		eloni 6396	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
13	12	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
14	13	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴)
15	11, 14	jca 511	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴))
16		ordelsuc 7840	. . . . . . . 8 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o ⊆ 𝐴))
17	16	biimpd 229	. . . . . . 7 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 → suc 1o ⊆ 𝐴))
18	15, 11, 17	sylc 65	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → suc 1o ⊆ 𝐴)
19	10, 18	eqsstrd 4034	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴)
20		omwordi 8608	. . . . 5 ⊢ ((2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ⊆ 𝐴 → (𝐵 ·o 2o) ⊆ (𝐵 ·o 𝐴)))
21	8, 19, 20	sylc 65	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 ·o 2o) ⊆ (𝐵 ·o 𝐴))
22	2, 21	eqsstrd 4034	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) ⊆ (𝐵 ·o 𝐴))
23	6, 6	jca 511	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐵 ∈ On))
24		simpr 484	. . . 4 ⊢ ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
25		oaordi 8583	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)))
26	25	imp 406	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
27	23, 24, 26	syl2an 596	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
28	22, 27	sseldd 3996	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴))
29	28	ex 412	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  Ord word 6385  Oncon0 6386  suc csuc 6388  (class class class)co 7431  1_oc1o 8498  2_oc2o 8499   +_o coa 8502   ·_o comu 8503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510

This theorem is referenced by: (None)