Theorem oaltom 44349

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 8572	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
2	1	ad2antlr 740	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
3		2on 8468	. . . . . . . 8 ⊢ 2o ∈ On
4	3	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 2o ∈ On)
5		simpl 488	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6		simpr 490	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
7	4, 5, 6	3jca 1146	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
8	7	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (2o ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On))
9		df-2o 8455	. . . . . . 7 ⊢ 2o = suc 1o
10	9	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc 1o)
11		simprl 783	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴)
12		eloni 6361	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → Ord 𝐴)
13	12	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord 𝐴)
14	13	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → Ord 𝐴)
15	11, 14	jca 521	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (1o ∈ 𝐴 ∧ Ord 𝐴))
16		ordelsuc 7814	. . . . . . . 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 3964	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴)
20		omwordi 8557	. . . . 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 3964	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) ⊆ (𝐵 ·o 𝐴))
23	6, 6	jca 521	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ∈ On ∧ 𝐵 ∈ On))
24		simpr 490	. . . 4 ⊢ ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵)
25		oaordi 8532	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵)))
26	25	imp 412	. . . 4 ⊢ (((𝐵 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
27	23, 24, 26	syl2an 608	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
28	22, 27	sseldd 3931	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴))
29	28	ex 418	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐵 +o 𝐴) ∈ (𝐵 ·o 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3898  Ord word 6350  Oncon0 6351  suc csuc 6353  (class class class)co 7408  1_oc1o 8447  2_oc2o 8448   +_o coa 8451   ·_o comu 8452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-oadd 8458  df-omul 8459

This theorem is used by: (None)