Theorem om0r 3142

Description: Ordinal multiplication with zero. Proposition 8.18(1) of [TakeutiZaring] p. 63.

Assertion

Ref	Expression
om0r	⊢ (A ∈ On → (∅ ·o A) = ∅)

Proof of Theorem om0r

Step	Hyp	Ref	Expression
1		opreq2 3007	. . 3 ⊢ (x = ∅ → (∅ ·o x) = (∅ ·o ∅))
2	1	cleq1d 1109	. 2 ⊢ (x = ∅ → ((∅ ·o x) = ∅ ↔ (∅ ·o ∅) = ∅))
3		opreq2 3007	. . 3 ⊢ (x = y → (∅ ·o x) = (∅ ·o y))
4	3	cleq1d 1109	. 2 ⊢ (x = y → ((∅ ·o x) = ∅ ↔ (∅ ·o y) = ∅))
5		opreq2 3007	. . 3 ⊢ (x = suc y → (∅ ·o x) = (∅ ·o suc y))
6	5	cleq1d 1109	. 2 ⊢ (x = suc y → ((∅ ·o x) = ∅ ↔ (∅ ·o suc y) = ∅))
7		opreq2 3007	. . 3 ⊢ (x = A → (∅ ·o x) = (∅ ·o A))
8	7	cleq1d 1109	. 2 ⊢ (x = A → ((∅ ·o x) = ∅ ↔ (∅ ·o A) = ∅))
9		om0x 3126	. 2 ⊢ (∅ ·o ∅) = ∅
10		0elon 2277	. . . . 5 ⊢ ∅ ∈ On
11		omsuc 3133	. . . . 5 ⊢ ((∅ ∈ On ∧ y ∈ On) → (∅ ·o suc y) = ((∅ ·o y) +o ∅))
12	10, 11	mpan 518	. . . 4 ⊢ (y ∈ On → (∅ ·o suc y) = ((∅ ·o y) +o ∅))
13		oa0 3124	. . . . . . 7 ⊢ (∅ ∈ On → (∅ +o ∅) = ∅)
14	10, 13	ax-mp 6	. . . . . 6 ⊢ (∅ +o ∅) = ∅
15	14	cleqcomi 1105	. . . . 5 ⊢ ∅ = (∅ +o ∅)
16	15	a1i 7	. . . 4 ⊢ (y ∈ On → ∅ = (∅ +o ∅))
17	12, 16	cleq12d 1115	. . 3 ⊢ (y ∈ On → ((∅ ·o suc y) = ∅ ↔ ((∅ ·o y) +o ∅) = (∅ +o ∅)))
18		opreq1 3006	. . 3 ⊢ ((∅ ·o y) = ∅ → ((∅ ·o y) +o ∅) = (∅ +o ∅))
19	17, 18	syl5bir 184	. 2 ⊢ (y ∈ On → ((∅ ·o y) = ∅ → (∅ ·o suc y) = ∅))
20		visset 1350	. . . . 5 ⊢ x ∈ V
21		omlim 3136	. . . . . 6 ⊢ ((∅ ∈ On ∧ (x ∈ V ∧ Lim x)) → (∅ ·o x) = ∪y ∈ x (∅ ·o y))
22	10, 21	mpan 518	. . . . 5 ⊢ ((x ∈ V ∧ Lim x) → (∅ ·o x) = ∪y ∈ x (∅ ·o y))
23	20, 22	mpan 518	. . . 4 ⊢ (Lim x → (∅ ·o x) = ∪y ∈ x (∅ ·o y))
24	23	cleq1d 1109	. . 3 ⊢ (Lim x → ((∅ ·o x) = ∅ ↔ ∪y ∈ x (∅ ·o y) = ∅))
25		iuneq2 2006	. . . 4 ⊢ (∀y ∈ x (∅ ·o y) = ∅ → ∪y ∈ x (∅ ·o y) = ∪y ∈ x ∅)
26		iun0 2028	. . . 4 ⊢ ∪y ∈ x ∅ = ∅
27	25, 26	syl6eq 1140	. . 3 ⊢ (∀y ∈ x (∅ ·o y) = ∅ → ∪y ∈ x (∅ ·o y) = ∅)
28	24, 27	syl5bir 184	. 2 ⊢ (Lim x → (∀y ∈ x (∅ ·o y) = ∅ → (∅ ·o x) = ∅))
29	2, 4, 6, 8, 9, 19, 28	tfinds 2401	1 ⊢ (A ∈ On → (∅ ·o A) = ∅)

Syntax hints:   → wi 2   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348  ∅c0 1707  ∪ciun 1994  Oncon0 2199  Lim wlim 2200  suc csuc 2201  (class class class)co 3001   +_o coa 3101   ·_o comu 3102

This theorem is referenced by:  nnm0r 3171

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106  df-omul 3107

metamath.org