Theorem oe1m 8480

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 7375	. . 3 ⊢ (𝑥 = ∅ → (1o ↑o 𝑥) = (1o ↑o ∅))
2	1	eqeq1d 2738	. 2 ⊢ (𝑥 = ∅ → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o ∅) = 1o))
3		oveq2 7375	. . 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 7375	. . 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 7375	. . 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 8417	. . 3 ⊢ 1o ∈ On
10		oe0 8457	. . 3 ⊢ (1o ∈ On → (1o ↑o ∅) = 1o)
11	9, 10	ax-mp 5	. 2 ⊢ (1o ↑o ∅) = 1o
12		oesuc 8462	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
13	9, 12	mpan 691	. . . 4 ⊢ (𝑦 ∈ On → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
14		oveq1 7374	. . . . 5 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = (1o ·o 1o))
15		om1 8477	. . . . . 6 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
16	9, 15	ax-mp 5	. . . . 5 ⊢ (1o ·o 1o) = 1o
17	14, 16	eqtrdi 2787	. . . 4 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = 1o)
18	13, 17	sylan9eq 2791	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ↑o 𝑦) = 1o) → (1o ↑o suc 𝑦) = 1o)
19	18	ex 412	. 2 ⊢ (𝑦 ∈ On → ((1o ↑o 𝑦) = 1o → (1o ↑o suc 𝑦) = 1o))
20		iuneq2 4953	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o)
21		vex 3433	. . . . . 6 ⊢ 𝑥 ∈ V
22		0lt1o 8439	. . . . . . . 8 ⊢ ∅ ∈ 1o
23		oelim 8469	. . . . . . . 8 ⊢ (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
24	22, 23	mpan2 692	. . . . . . 7 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
25	9, 24	mpan 691	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
26	21, 25	mpan 691	. . . . 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 6387	. . . . . 6 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		ne0i 4281	. . . . . 6 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
30		iunconst 4943	. . . . . 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 282	. . 3 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o))
34	20, 33	imbitrrid 246	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → (1o ↑o 𝑥) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 7811	1 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  Vcvv 3429  ∅c0 4273  ∪ ciun 4933  Oncon0 6323  Lim wlim 6324  suc csuc 6325  (class class class)co 7367  1_oc1o 8398   ·_o comu 8403   ↑_o coe 8404

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410  df-oexp 8411

This theorem is referenced by:  oewordi  8527  oeoe  8535  cantnflem2  9611