Theorem oe1m 8171

Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)

Assertion

Ref	Expression
oe1m	⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)

Proof of Theorem oe1m

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7164	. . 3 ⊢ (𝑥 = ∅ → (1o ↑o 𝑥) = (1o ↑o ∅))
2	1	eqeq1d 2823	. 2 ⊢ (𝑥 = ∅ → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o ∅) = 1o))
3		oveq2 7164	. . 3 ⊢ (𝑥 = 𝑦 → (1o ↑o 𝑥) = (1o ↑o 𝑦))
4	3	eqeq1d 2823	. 2 ⊢ (𝑥 = 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝑦) = 1o))
5		oveq2 7164	. . 3 ⊢ (𝑥 = suc 𝑦 → (1o ↑o 𝑥) = (1o ↑o suc 𝑦))
6	5	eqeq1d 2823	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o suc 𝑦) = 1o))
7		oveq2 7164	. . 3 ⊢ (𝑥 = 𝐴 → (1o ↑o 𝑥) = (1o ↑o 𝐴))
8	7	eqeq1d 2823	. 2 ⊢ (𝑥 = 𝐴 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝐴) = 1o))
9		1on 8109	. . 3 ⊢ 1o ∈ On
10		oe0 8147	. . 3 ⊢ (1o ∈ On → (1o ↑o ∅) = 1o)
11	9, 10	ax-mp 5	. 2 ⊢ (1o ↑o ∅) = 1o
12		oesuc 8152	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
13	9, 12	mpan 688	. . . 4 ⊢ (𝑦 ∈ On → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
14		oveq1 7163	. . . . 5 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = (1o ·o 1o))
15		om1 8168	. . . . . 6 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
16	9, 15	ax-mp 5	. . . . 5 ⊢ (1o ·o 1o) = 1o
17	14, 16	syl6eq 2872	. . . 4 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = 1o)
18	13, 17	sylan9eq 2876	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ↑o 𝑦) = 1o) → (1o ↑o suc 𝑦) = 1o)
19	18	ex 415	. 2 ⊢ (𝑦 ∈ On → ((1o ↑o 𝑦) = 1o → (1o ↑o suc 𝑦) = 1o))
20		iuneq2 4938	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o)
21		vex 3497	. . . . . 6 ⊢ 𝑥 ∈ V
22		0lt1o 8129	. . . . . . . 8 ⊢ ∅ ∈ 1o
23		oelim 8159	. . . . . . . 8 ⊢ (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
24	22, 23	mpan2 689	. . . . . . 7 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
25	9, 24	mpan 688	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
26	21, 25	mpan 688	. . . . 5 ⊢ (Lim 𝑥 → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
27	26	eqeq1d 2823	. . . 4 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
28		0ellim 6253	. . . . . 6 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		ne0i 4300	. . . . . 6 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
30		iunconst 4928	. . . . . 6 ⊢ (𝑥 ≠ ∅ → ∪ 𝑦 ∈ 𝑥 1o = 1o)
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (Lim 𝑥 → ∪ 𝑦 ∈ 𝑥 1o = 1o)
32	31	eqeq2d 2832	. . . 4 ⊢ (Lim 𝑥 → (∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
33	27, 32	bitr4d 284	. . 3 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o))
34	20, 33	syl5ibr 248	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → (1o ↑o 𝑥) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 7574	1 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494  ∅c0 4291  ∪ ciun 4919  Oncon0 6191  Lim wlim 6192  suc csuc 6193  (class class class)co 7156  1_oc1o 8095   ·_o comu 8100   ↑_o coe 8101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-omul 8107  df-oexp 8108

This theorem is referenced by:  oewordi  8217  oeoe  8225  cantnflem2  9153