Theorem oe1m 7911

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 6932	. . 3 ⊢ (𝑥 = ∅ → (1o ↑o 𝑥) = (1o ↑o ∅))
2	1	eqeq1d 2780	. 2 ⊢ (𝑥 = ∅ → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o ∅) = 1o))
3		oveq2 6932	. . 3 ⊢ (𝑥 = 𝑦 → (1o ↑o 𝑥) = (1o ↑o 𝑦))
4	3	eqeq1d 2780	. 2 ⊢ (𝑥 = 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝑦) = 1o))
5		oveq2 6932	. . 3 ⊢ (𝑥 = suc 𝑦 → (1o ↑o 𝑥) = (1o ↑o suc 𝑦))
6	5	eqeq1d 2780	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o suc 𝑦) = 1o))
7		oveq2 6932	. . 3 ⊢ (𝑥 = 𝐴 → (1o ↑o 𝑥) = (1o ↑o 𝐴))
8	7	eqeq1d 2780	. 2 ⊢ (𝑥 = 𝐴 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝐴) = 1o))
9		1on 7852	. . 3 ⊢ 1o ∈ On
10		oe0 7888	. . 3 ⊢ (1o ∈ On → (1o ↑o ∅) = 1o)
11	9, 10	ax-mp 5	. 2 ⊢ (1o ↑o ∅) = 1o
12		oesuc 7893	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
13	9, 12	mpan 680	. . . 4 ⊢ (𝑦 ∈ On → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
14		oveq1 6931	. . . . 5 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = (1o ·o 1o))
15		om1 7908	. . . . . 6 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
16	9, 15	ax-mp 5	. . . . 5 ⊢ (1o ·o 1o) = 1o
17	14, 16	syl6eq 2830	. . . 4 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = 1o)
18	13, 17	sylan9eq 2834	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ↑o 𝑦) = 1o) → (1o ↑o suc 𝑦) = 1o)
19	18	ex 403	. 2 ⊢ (𝑦 ∈ On → ((1o ↑o 𝑦) = 1o → (1o ↑o suc 𝑦) = 1o))
20		iuneq2 4772	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o)
21		vex 3401	. . . . . 6 ⊢ 𝑥 ∈ V
22		0lt1o 7870	. . . . . . . 8 ⊢ ∅ ∈ 1o
23		oelim 7900	. . . . . . . 8 ⊢ (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
24	22, 23	mpan2 681	. . . . . . 7 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
25	9, 24	mpan 680	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
26	21, 25	mpan 680	. . . . 5 ⊢ (Lim 𝑥 → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
27	26	eqeq1d 2780	. . . 4 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
28		0ellim 6040	. . . . . 6 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		ne0i 4149	. . . . . 6 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
30		iunconst 4764	. . . . . 6 ⊢ (𝑥 ≠ ∅ → ∪ 𝑦 ∈ 𝑥 1o = 1o)
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (Lim 𝑥 → ∪ 𝑦 ∈ 𝑥 1o = 1o)
32	31	eqeq2d 2788	. . . 4 ⊢ (Lim 𝑥 → (∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
33	27, 32	bitr4d 274	. . 3 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o))
34	20, 33	syl5ibr 238	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → (1o ↑o 𝑥) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 7339	1 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  ∀wral 3090  Vcvv 3398  ∅c0 4141  ∪ ciun 4755  Oncon0 5978  Lim wlim 5979  suc csuc 5980  (class class class)co 6924  1_oc1o 7838   ·_o comu 7843   ↑_o coe 7844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-omul 7850  df-oexp 7851

This theorem is referenced by:  oewordi  7957  oeoe  7965  cantnflem2  8886