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Theorem cantnflem3 8471

Description: Lemma for cantnf 8473. 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 7570 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 8461 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 = (1^st ‘𝑃)
cantnf.z	⊢ 𝑍 = (2^nd ‘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 8470	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)))
10		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = 𝑋
11		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = 𝑌
12		eqid 2610	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = 𝑍
13	10, 11, 12	3pm3.2i 1232	. . . . . . . . . . . . . 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 = (1^st ‘𝑃)
17		cantnf.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (2^nd ‘𝑃)
18	14, 15, 16, 17	oeeui 7569	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)) → (((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶) ↔ (𝑋 = 𝑋 ∧ 𝑌 = 𝑌 ∧ 𝑍 = 𝑍)))
19	13, 18	mpbiri 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)) → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶))
20	9, 19	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) ∧ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶))
21	20	simpld 474	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)))
22	21	simp1d 1066	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ On)
23		oecl 7504	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_𝑜 𝑋) ∈ On)
24	2, 22, 23	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ∈ On)
25	21	simp2d 1067	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1_𝑜))
26	25	eldifad 3552	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
27		onelon 5665	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
28	2, 26, 27	syl2anc 691	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ On)
29		dif1o 7467	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (𝐴 ∖ 1_𝑜) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅))
30	29	simprbi 479	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐴 ∖ 1_𝑜) → 𝑌 ≠ ∅)
31	25, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ≠ ∅)
32		on0eln0 5697	. . . . . . . . . . 11 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
33	28, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
34	31, 33	mpbird 246	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ 𝑌)
35		omword1 7540	. . . . . . . . 9 ⊢ ((((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑_𝑜 𝑋) ⊆ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
36	24, 28, 34, 35	syl21anc 1317	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ⊆ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
37		omcl 7503	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ∈ On)
38	24, 28, 37	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ∈ On)
39	21	simp3d 1068	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑_𝑜 𝑋))
40		onelon 5665	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) → 𝑍 ∈ On)
41	24, 39, 40	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ On)
42		oaword1 7519	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ⊆ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍))
44	20	simprd 478	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶)
45	43, 44	sseqtrd 3604	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ⊆ 𝐶)
46	36, 45	sstrd 3578	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ⊆ 𝐶)
47		oecl 7504	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_𝑜 𝐵) ∈ On)
48	2, 3, 47	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ∈ On)
49		ontr2 5689	. . . . . . . 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 691	. . . . . . 7 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 ↑_𝑜 𝐵)) → (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵)))
51	46, 6, 50	mp2and 711	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵))
52	9	simpld 474	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2_𝑜))
53		oeord 7555	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2_𝑜)) → (𝑋 ∈ 𝐵 ↔ (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵)))
54	22, 3, 52, 53	syl3anc 1318	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵)))
55	51, 54	mpbird 246	. . . . 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 7195	. . . . . . . . . . . . . . 15 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
59	1, 2, 3	cantnfs 8446	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
60	4, 59	mpbid 221	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
61	60	simpld 474	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
62		fdm 5964	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐵⟶𝐴 → dom 𝐺 = 𝐵)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐺 = 𝐵)
64	58, 63	syl5sseq 3616	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
65	64	sselda 3568	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ 𝐵)
66		onelon 5665	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
67	57, 65, 66	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
68		oecl 7504	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑_𝑜 𝑥) ∈ On)
69	56, 67, 68	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ∈ On)
70	61	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵⟶𝐴)
71	70, 65	ffvelrnd 6268	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ∈ 𝐴)
72		onelon 5665	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑥) ∈ 𝐴) → (𝐺‘𝑥) ∈ On)
73	56, 71, 72	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ∈ On)
74		ffn 5958	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐵⟶𝐴 → 𝐺 Fn 𝐵)
75	61, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 Fn 𝐵)
76		0ex 4718	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
77	76	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ∈ V)
78		elsuppfn 7190	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥 ∈ 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)))
79	75, 3, 77, 78	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥 ∈ 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)))
80	79	simplbda 652	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ≠ ∅)
81		on0eln0 5697	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑥) ∈ On → (∅ ∈ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ ∅))
82	73, 81	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (∅ ∈ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ ∅))
83	80, 82	mpbird 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺‘𝑥))
84		omword1 7540	. . . . . . . . . . 11 ⊢ ((((𝐴 ↑_𝑜 𝑥) ∈ On ∧ (𝐺‘𝑥) ∈ On) ∧ ∅ ∈ (𝐺‘𝑥)) → (𝐴 ↑_𝑜 𝑥) ⊆ ((𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐺‘𝑥)))
85	69, 73, 83, 84	syl21anc 1317	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ⊆ ((𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐺‘𝑥)))
86		eqid 2610	. . . . . . . . . . . 12 ⊢ OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
87	4	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐺 ∈ 𝑆)
88		eqid 2610	. . . . . . . . . . . 12 ⊢ seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·_𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +_𝑜 𝑧)), ∅) = seq_𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·_𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +_𝑜 𝑧)), ∅)
89	1, 56, 57, 86, 87, 88, 65	cantnfle 8451	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐺‘𝑥)) ⊆ ((𝐴 CNF 𝐵)‘𝐺))
90		cantnf.v	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
91	90	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
92	89, 91	sseqtrd 3604	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐺‘𝑥)) ⊆ 𝑍)
93	85, 92	sstrd 3578	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ⊆ 𝑍)
94	39	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴 ↑_𝑜 𝑋))
95	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑋) ∈ On)
96		ontr2 5689	. . . . . . . . . 10 ⊢ (((𝐴 ↑_𝑜 𝑥) ∈ On ∧ (𝐴 ↑_𝑜 𝑋) ∈ On) → (((𝐴 ↑_𝑜 𝑥) ⊆ 𝑍 ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) → (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
97	69, 95, 96	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (((𝐴 ↑_𝑜 𝑥) ⊆ 𝑍 ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) → (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
98	93, 94, 97	mp2and 711	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋))
99	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
100	52	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2_𝑜))
101		oeord 7555	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2_𝑜)) → (𝑥 ∈ 𝑋 ↔ (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
102	67, 99, 100, 101	syl3anc 1318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝑥 ∈ 𝑋 ↔ (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
103	98, 102	mpbird 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ 𝑋)
104	103	ex 449	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥 ∈ 𝑋))
105	104	ssrdv 3574	. . . . 5 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
106		cantnf.f	. . . . 5 ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))
107	1, 2, 3, 4, 55, 26, 105, 106	cantnfp1 8461	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
108	107	simprd 478	. . 3 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
109	90	oveq2d 6565	. . 3 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍))
110	108, 109, 44	3eqtrd 2648	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
111	1, 2, 3	cantnff 8454	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
112		ffn 5958	. . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵) → (𝐴 CNF 𝐵) Fn 𝑆)
113	111, 112	syl 17	. . 3 ⊢ (𝜑 → (𝐴 CNF 𝐵) Fn 𝑆)
114	107	simpld 474	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
115		fnfvelrn 6264	. . 3 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ 𝐹 ∈ 𝑆) → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
116	113, 114, 115	syl2anc 691	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
117	110, 116	eqeltrrd 2689	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 ifcif 4036 ⟨cop 4131 ∪ cuni 4372 ∩ cint 4410 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 E cep 4947 dom cdm 5038 ran crn 5039 Oncon0 5640 ℩cio 5766 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 supp csupp 7182 seq_𝜔cseqom 7429 1_𝑜c1o 7440 2_𝑜c2o 7441 +_𝑜 coa 7444 ·_𝑜 comu 7445 ↑_𝑜 coe 7446 finSupp cfsupp 8158 OrdIsocoi 8297 CNF ccnf 8441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-cnf 8442

This theorem is referenced by: cantnflem4 8472

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