Theorem oe1m 8492

Description: Ordinal exponentiation with a base of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. Lemma 2.17 of [Schloeder] p. 6. (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 7365	. . 3 ⊢ (𝑥 = ∅ → (1o ↑o 𝑥) = (1o ↑o ∅))
2	1	eqeq1d 2738	. 2 ⊢ (𝑥 = ∅ → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o ∅) = 1o))
3		oveq2 7365	. . 3 ⊢ (𝑥 = 𝑦 → (1o ↑o 𝑥) = (1o ↑o 𝑦))
4	3	eqeq1d 2738	. 2 ⊢ (𝑥 = 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝑦) = 1o))
5		oveq2 7365	. . 3 ⊢ (𝑥 = suc 𝑦 → (1o ↑o 𝑥) = (1o ↑o suc 𝑦))
6	5	eqeq1d 2738	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o suc 𝑦) = 1o))
7		oveq2 7365	. . 3 ⊢ (𝑥 = 𝐴 → (1o ↑o 𝑥) = (1o ↑o 𝐴))
8	7	eqeq1d 2738	. 2 ⊢ (𝑥 = 𝐴 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝐴) = 1o))
9		1on 8424	. . 3 ⊢ 1o ∈ On
10		oe0 8468	. . 3 ⊢ (1o ∈ On → (1o ↑o ∅) = 1o)
11	9, 10	ax-mp 5	. 2 ⊢ (1o ↑o ∅) = 1o
12		oesuc 8473	. . . . 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 7364	. . . . 5 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = (1o ·o 1o))
15		om1 8489	. . . . . 6 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
16	9, 15	ax-mp 5	. . . . 5 ⊢ (1o ·o 1o) = 1o
17	14, 16	eqtrdi 2792	. . . 4 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = 1o)
18	13, 17	sylan9eq 2796	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ↑o 𝑦) = 1o) → (1o ↑o suc 𝑦) = 1o)
19	18	ex 413	. 2 ⊢ (𝑦 ∈ On → ((1o ↑o 𝑦) = 1o → (1o ↑o suc 𝑦) = 1o))
20		iuneq2 4973	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o)
21		vex 3449	. . . . . 6 ⊢ 𝑥 ∈ V
22		0lt1o 8450	. . . . . . . 8 ⊢ ∅ ∈ 1o
23		oelim 8480	. . . . . . . 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 2738	. . . 4 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
28		0ellim 6380	. . . . . 6 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		ne0i 4294	. . . . . 6 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
30		iunconst 4963	. . . . . 6 ⊢ (𝑥 ≠ ∅ → ∪ 𝑦 ∈ 𝑥 1o = 1o)
31	28, 29, 30	3syl 18	. . . . 5 ⊢ (Lim 𝑥 → ∪ 𝑦 ∈ 𝑥 1o = 1o)
32	31	eqeq2d 2747	. . . 4 ⊢ (Lim 𝑥 → (∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
33	27, 32	bitr4d 281	. . 3 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o))
34	20, 33	syl5ibr 245	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → (1o ↑o 𝑥) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 7796	1 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3445  ∅c0 4282  ∪ ciun 4954  Oncon0 6317  Lim wlim 6318  suc csuc 6319  (class class class)co 7357  1_oc1o 8405   ·_o comu 8410   ↑_o coe 8411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-oexp 8418

This theorem is referenced by:  oewordi  8538  oeoe  8546  cantnflem2  9626