Theorem cantnflem3 9732

Description: Lemma for cantnf 9734. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 8642 to factor 𝐶 into the form ((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑_o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑_o 𝑋) ≤ (𝐴 ↑_o 𝑋) ·_o 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑_o 𝑋) ·_o 𝑌 using cantnfp1 9722 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
cantnf.c	⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
cantnf.s	⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e	⊢ (𝜑 → ∅ ∈ 𝐶)
cantnf.x	⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}
cantnf.p	⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
cantnf.y	⊢ 𝑌 = (1st ‘𝑃)
cantnf.z	⊢ 𝑍 = (2nd ‘𝑃)
cantnf.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
cantnf.v	⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
cantnf.f	⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))

Assertion

Ref	Expression
cantnflem3	⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Distinct variable groups:   𝑡,𝑐,𝑤,𝑥,𝑦,𝑧,𝐵   𝑎,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧,𝐶   𝑡,𝑎,𝐴,𝑏,𝑐,𝑑,𝑤,𝑥,𝑦,𝑧   𝑇,𝑐,𝑡   𝑤,𝐹,𝑥,𝑦,𝑧   𝑆,𝑐,𝑡,𝑥,𝑦,𝑧   𝑡,𝑍,𝑥,𝑦,𝑧   𝐺,𝑐,𝑡,𝑤,𝑥,𝑦,𝑧   𝜑,𝑡,𝑥,𝑦,𝑧   𝑡,𝑌,𝑤,𝑥,𝑦,𝑧   𝑋,𝑎,𝑏,𝑑,𝑡,𝑤,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑤,𝑎,𝑏,𝑐,𝑑)   𝐵(𝑎,𝑏,𝑑)   𝐶(𝑡)   𝑃(𝑥,𝑦,𝑧,𝑤,𝑡,𝑎,𝑏,𝑐,𝑑)   𝑆(𝑤,𝑎,𝑏,𝑑)   𝑇(𝑥,𝑦,𝑧,𝑤,𝑎,𝑏,𝑑)   𝐹(𝑡,𝑎,𝑏,𝑐,𝑑)   𝐺(𝑎,𝑏,𝑑)   𝑋(𝑐)   𝑌(𝑎,𝑏,𝑐,𝑑)   𝑍(𝑤,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem cantnflem3

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfs.s	. . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
2		cantnfs.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ On)
3		cantnfs.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ On)
4		cantnf.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
5		oemapval.t	. . . . . . . . . . . . . 14 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
6		cantnf.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
7		cantnf.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
8		cantnf.e	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ∈ 𝐶)
9	1, 2, 3, 5, 6, 7, 8	cantnflem2 9731	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
10		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = 𝑋
11		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = 𝑌
12		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = 𝑍
13	10, 11, 12	3pm3.2i 1339	. . . . . . . . . . . . . 14 ⊢ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)
14		cantnf.x	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}
15		cantnf.p	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))
16		cantnf.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = (1st ‘𝑃)
17		cantnf.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (2nd ‘𝑃)
18	14, 15, 16, 17	oeeui 8641	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶) ↔ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)))
19	13, 18	mpbiri 258	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
20	9, 19	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) ∧ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶))
21	20	simpld 494	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1o) ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)))
22	21	simp1d 1142	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ On)
23		oecl 8576	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑o 𝑋) ∈ On)
24	2, 22, 23	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ∈ On)
25	21	simp2d 1143	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1o))
26	25	eldifad 3962	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
27		onelon 6408	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
28	2, 26, 27	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ On)
29		dif1o 8539	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (𝐴 ∖ 1o) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅))
30	29	simprbi 496	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐴 ∖ 1o) → 𝑌 ≠ ∅)
31	25, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ≠ ∅)
32		on0eln0 6439	. . . . . . . . . . 11 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
33	28, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
34	31, 33	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ 𝑌)
35		omword1 8612	. . . . . . . . 9 ⊢ ((((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
36	24, 28, 34, 35	syl21anc 837	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ⊆ ((𝐴 ↑o 𝑋) ·o 𝑌))
37		omcl 8575	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
38	24, 28, 37	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On)
39	21	simp3d 1144	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑o 𝑋))
40		onelon 6408	. . . . . . . . . . 11 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) → 𝑍 ∈ On)
41	24, 39, 40	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ On)
42		oaword1 8591	. . . . . . . . . 10 ⊢ ((((𝐴 ↑o 𝑋) ·o 𝑌) ∈ On ∧ 𝑍 ∈ On) → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
43	38, 41, 42	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
44	20	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍) = 𝐶)
45	43, 44	sseqtrd 4019	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ↑o 𝑋) ·o 𝑌) ⊆ 𝐶)
46	36, 45	sstrd 3993	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ⊆ 𝐶)
47		oecl 8576	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
48	2, 3, 47	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On)
49		ontr2 6430	. . . . . . . 8 ⊢ (((𝐴 ↑o 𝑋) ∈ On ∧ (𝐴 ↑o 𝐵) ∈ On) → (((𝐴 ↑o 𝑋) ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → (𝐴 ↑o 𝑋) ∈ (𝐴 ↑o 𝐵)))
50	24, 48, 49	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (((𝐴 ↑o 𝑋) ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → (𝐴 ↑o 𝑋) ∈ (𝐴 ↑o 𝐵)))
51	46, 6, 50	mp2and 699	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑o 𝑋) ∈ (𝐴 ↑o 𝐵))
52	9	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o))
53		oeord 8627	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑋 ∈ 𝐵 ↔ (𝐴 ↑o 𝑋) ∈ (𝐴 ↑o 𝐵)))
54	22, 3, 52, 53	syl3anc 1372	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝐴 ↑o 𝑋) ∈ (𝐴 ↑o 𝐵)))
55	51, 54	mpbird 257	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
56	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ On)
57	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐵 ∈ On)
58		suppssdm 8203	. . . . . . . . . . . . . . 15 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
59	1, 2, 3	cantnfs 9707	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
60	4, 59	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
61	60	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
62	58, 61	fssdm 6754	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
63	62	sselda 3982	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ 𝐵)
64		onelon 6408	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
65	57, 63, 64	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
66		oecl 8576	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑o 𝑥) ∈ On)
67	56, 65, 66	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑o 𝑥) ∈ On)
68	61	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵⟶𝐴)
69	68, 63	ffvelcdmd 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ∈ 𝐴)
70		onelon 6408	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑥) ∈ 𝐴) → (𝐺‘𝑥) ∈ On)
71	56, 69, 70	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ∈ On)
72	61	ffnd 6736	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 Fn 𝐵)
73	8	elexd 3503	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ∈ V)
74		elsuppfn 8196	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥 ∈ 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)))
75	72, 3, 73, 74	syl3anc 1372	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥 ∈ 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)))
76	75	simplbda 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ≠ ∅)
77		on0eln0 6439	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑥) ∈ On → (∅ ∈ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ ∅))
78	71, 77	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (∅ ∈ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ ∅))
79	76, 78	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺‘𝑥))
80		omword1 8612	. . . . . . . . . . 11 ⊢ ((((𝐴 ↑o 𝑥) ∈ On ∧ (𝐺‘𝑥) ∈ On) ∧ ∅ ∈ (𝐺‘𝑥)) → (𝐴 ↑o 𝑥) ⊆ ((𝐴 ↑o 𝑥) ·o (𝐺‘𝑥)))
81	67, 71, 79, 80	syl21anc 837	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑o 𝑥) ⊆ ((𝐴 ↑o 𝑥) ·o (𝐺‘𝑥)))
82		eqid 2736	. . . . . . . . . . . 12 ⊢ OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
83	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐺 ∈ 𝑆)
84		eqid 2736	. . . . . . . . . . . 12 ⊢ seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·o (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +o 𝑧)), ∅) = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·o (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +o 𝑧)), ∅)
85	1, 56, 57, 82, 83, 84, 63	cantnfle 9712	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 ↑o 𝑥) ·o (𝐺‘𝑥)) ⊆ ((𝐴 CNF 𝐵)‘𝐺))
86		cantnf.v	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
87	86	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
88	85, 87	sseqtrd 4019	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 ↑o 𝑥) ·o (𝐺‘𝑥)) ⊆ 𝑍)
89	81, 88	sstrd 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑o 𝑥) ⊆ 𝑍)
90	39	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴 ↑o 𝑋))
91	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑o 𝑋) ∈ On)
92		ontr2 6430	. . . . . . . . . 10 ⊢ (((𝐴 ↑o 𝑥) ∈ On ∧ (𝐴 ↑o 𝑋) ∈ On) → (((𝐴 ↑o 𝑥) ⊆ 𝑍 ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) → (𝐴 ↑o 𝑥) ∈ (𝐴 ↑o 𝑋)))
93	67, 91, 92	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (((𝐴 ↑o 𝑥) ⊆ 𝑍 ∧ 𝑍 ∈ (𝐴 ↑o 𝑋)) → (𝐴 ↑o 𝑥) ∈ (𝐴 ↑o 𝑋)))
94	89, 90, 93	mp2and 699	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑o 𝑥) ∈ (𝐴 ↑o 𝑋))
95	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
96	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2o))
97		oeord 8627	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2o)) → (𝑥 ∈ 𝑋 ↔ (𝐴 ↑o 𝑥) ∈ (𝐴 ↑o 𝑋)))
98	65, 95, 96, 97	syl3anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝑥 ∈ 𝑋 ↔ (𝐴 ↑o 𝑥) ∈ (𝐴 ↑o 𝑋)))
99	94, 98	mpbird 257	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ 𝑋)
100	99	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥 ∈ 𝑋))
101	100	ssrdv 3988	. . . . 5 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
102		cantnf.f	. . . . 5 ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))
103	1, 2, 3, 4, 55, 26, 101, 102	cantnfp1 9722	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
104	103	simprd 495	. . 3 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
105	86	oveq2d 7448	. . 3 ⊢ (𝜑 → (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍))
106	104, 105, 44	3eqtrd 2780	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
107	1, 2, 3	cantnff 9715	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
108	107	ffnd 6736	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵) Fn 𝑆)
109	103	simpld 494	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
110		fnfvelrn 7099	. . 3 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ 𝐹 ∈ 𝑆) → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
111	108, 109, 110	syl2anc 584	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
112	106, 111	eqeltrrd 2841	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3435  Vcvv 3479   ∖ cdif 3947   ⊆ wss 3950  ∅c0 4332  ifcif 4524  ⟨cop 4631  ∪ cuni 4906  ∩ cint 4945   class class class wbr 5142  {copab 5204   ↦ cmpt 5224   E cep 5582  dom cdm 5684  ran crn 5685  Oncon0 6383  ℩cio 6511   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  1^st c1st 8013  2^nd c2nd 8014   supp csupp 8186  seq_ωcseqom 8488  1_oc1o 8500  2_oc2o 8501   +_o coa 8504   ·_o comu 8505   ↑_o coe 8506   finSupp cfsupp 9402  OrdIsocoi 9550   CNF ccnf 9702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-seqom 8489  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-oi 9551  df-cnf 9703

This theorem is referenced by:  cantnflem4  9733