Theorem oe1m 8561

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 7432	. . 3 ⊢ (𝑥 = ∅ → (1o ↑o 𝑥) = (1o ↑o ∅))
2	1	eqeq1d 2762	. 2 ⊢ (𝑥 = ∅ → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o ∅) = 1o))
3		oveq2 7432	. . 3 ⊢ (𝑥 = 𝑦 → (1o ↑o 𝑥) = (1o ↑o 𝑦))
4	3	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝑦) = 1o))
5		oveq2 7432	. . 3 ⊢ (𝑥 = suc 𝑦 → (1o ↑o 𝑥) = (1o ↑o suc 𝑦))
6	5	eqeq1d 2762	. 2 ⊢ (𝑥 = suc 𝑦 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o suc 𝑦) = 1o))
7		oveq2 7432	. . 3 ⊢ (𝑥 = 𝐴 → (1o ↑o 𝑥) = (1o ↑o 𝐴))
8	7	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝐴 → ((1o ↑o 𝑥) = 1o ↔ (1o ↑o 𝐴) = 1o))
9		1on 8497	. . 3 ⊢ 1o ∈ On
10		oe0 8538	. . 3 ⊢ (1o ∈ On → (1o ↑o ∅) = 1o)
11	9, 10	ax-mp 5	. 2 ⊢ (1o ↑o ∅) = 1o
12		oesuc 8543	. . . . 5 ⊢ ((1o ∈ On ∧ 𝑦 ∈ On) → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
13	9, 12	mpan 703	. . . 4 ⊢ (𝑦 ∈ On → (1o ↑o suc 𝑦) = ((1o ↑o 𝑦) ·o 1o))
14		oveq1 7431	. . . . 5 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = (1o ·o 1o))
15		om1 8558	. . . . . 6 ⊢ (1o ∈ On → (1o ·o 1o) = 1o)
16	9, 15	ax-mp 5	. . . . 5 ⊢ (1o ·o 1o) = 1o
17	14, 16	eqtrdi 2811	. . . 4 ⊢ ((1o ↑o 𝑦) = 1o → ((1o ↑o 𝑦) ·o 1o) = 1o)
18	13, 17	sylan9eq 2815	. . 3 ⊢ ((𝑦 ∈ On ∧ (1o ↑o 𝑦) = 1o) → (1o ↑o suc 𝑦) = 1o)
19	18	ex 418	. 2 ⊢ (𝑦 ∈ On → ((1o ↑o 𝑦) = 1o → (1o ↑o suc 𝑦) = 1o))
20		iuneq2 4970	. . 3 ⊢ (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o)
21		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
22		0lt1o 8520	. . . . . . . 8 ⊢ ∅ ∈ 1o
23		oelim 8550	. . . . . . . 8 ⊢ (((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) ∧ ∅ ∈ 1o) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
24	22, 23	mpan2 704	. . . . . . 7 ⊢ ((1o ∈ On ∧ (𝑥 ∈ V ∧ Lim 𝑥)) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
25	9, 24	mpan 703	. . . . . 6 ⊢ ((𝑥 ∈ V ∧ Lim 𝑥) → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
26	21, 25	mpan 703	. . . . 5 ⊢ (Lim 𝑥 → (1o ↑o 𝑥) = ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦))
27	26	eqeq1d 2762	. . . 4 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
28		0ellim 6430	. . . . . 6 ⊢ (Lim 𝑥 → ∅ ∈ 𝑥)
29		ne0i 4286	. . . . . 6 ⊢ (∅ ∈ 𝑥 → 𝑥 ≠ ∅)
30		iunconst 4960	. . . . . 6 ⊢ (𝑥 ≠ ∅ → ∪ 𝑦 ∈ 𝑥 1o = 1o)
31	28, 29, 30	3syl 19	. . . . 5 ⊢ (Lim 𝑥 → ∪ 𝑦 ∈ 𝑥 1o = 1o)
32	31	eqeq2d 2771	. . . 4 ⊢ (Lim 𝑥 → (∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o))
33	27, 32	bitr4d 285	. . 3 ⊢ (Lim 𝑥 → ((1o ↑o 𝑥) = 1o ↔ ∪ 𝑦 ∈ 𝑥 (1o ↑o 𝑦) = ∪ 𝑦 ∈ 𝑥 1o))
34	20, 33	imbitrrid 249	. 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (1o ↑o 𝑦) = 1o → (1o ↑o 𝑥) = 1o))
35	2, 4, 6, 8, 11, 19, 34	tfinds 7875	1 ⊢ (𝐴 ∈ On → (1o ↑o 𝐴) = 1o)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4277  ∪ ciun 4950  Oncon0 6365  Lim wlim 6366  suc csuc 6367  (class class class)co 7424  1_oc1o 8477   ·_o comu 8482   ↑_o coe 8483

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-1o 8484  df-oadd 8488  df-omul 8489  df-oexp 8490

This theorem is used by:  oewordi  8608  oeoe  8616  cantnflem2  9698