Theorem oaltom 42758

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 42757	. . . . 5 ⊢ (𝐵 ∈ On → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
2	1	ad2antlr 726	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐵) = (𝐵 ·o 2o))
3		2on 8494	. . . . . . . 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 1126	. . . . . 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 8481	. . . . . . 7 ⊢ 2o = suc 1o
10	9	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o = suc 1o)
11		simprl 770	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 1o ∈ 𝐴)
12		eloni 6373	. . . . . . . . . 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 7817	. . . . . . . 8 ⊢ ((1o ∈ 𝐴 ∧ Ord 𝐴) → (1o ∈ 𝐴 ↔ suc 1o ⊆ 𝐴))
17	16	biimpd 228	. . . . . . 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 4016	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → 2o ⊆ 𝐴)
20		omwordi 8585	. . . . 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 4016	. . 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 8560	. . . . 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 595	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (1o ∈ 𝐴 ∧ 𝐴 ∈ 𝐵)) → (𝐵 +o 𝐴) ∈ (𝐵 +o 𝐵))
28	22, 27	sseldd 3979	. 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 1085   = wceq 1534   ∈ wcel 2099   ⊆ wss 3944  Ord word 6362  Oncon0 6363  suc csuc 6365  (class class class)co 7414  1_oc1o 8473  2_oc2o 8474   +_o coa 8477   ·_o comu 8478

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

This theorem is referenced by: (None)