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Theorem cantnflem3 8588

Description: Lemma for cantnf 8590. 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 7683 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 8578 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 8587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2_𝑜) ∧ 𝐶 ∈ (On ∖ 1_𝑜)))
10		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = 𝑋
11		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = 𝑌
12		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = 𝑍
13	10, 11, 12	3pm3.2i 1239	. . . . . . . . . . . . . 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 7682	. . . . . . . . . . . . . 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 248	. . . . . . . . . . . . 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 475	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑋 ∈ On ∧ 𝑌 ∈ (𝐴 ∖ 1_𝑜) ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)))
22	21	simp1d 1073	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ On)
23		oecl 7617	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_𝑜 𝑋) ∈ On)
24	2, 22, 23	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ∈ On)
25	21	simp2d 1074	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ (𝐴 ∖ 1_𝑜))
26	25	eldifad 3586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
27		onelon 5748	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
28	2, 26, 27	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ On)
29		dif1o 7580	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ (𝐴 ∖ 1_𝑜) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ≠ ∅))
30	29	simprbi 480	. . . . . . . . . . 11 ⊢ (𝑌 ∈ (𝐴 ∖ 1_𝑜) → 𝑌 ≠ ∅)
31	25, 30	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ≠ ∅)
32		on0eln0 5780	. . . . . . . . . . 11 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
33	28, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
34	31, 33	mpbird 247	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ 𝑌)
35		omword1 7653	. . . . . . . . 9 ⊢ ((((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) ∧ ∅ ∈ 𝑌) → (𝐴 ↑_𝑜 𝑋) ⊆ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
36	24, 28, 34, 35	syl21anc 1325	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ⊆ ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌))
37		omcl 7616	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑌 ∈ On) → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ∈ On)
38	24, 28, 37	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ∈ On)
39	21	simp3d 1075	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ (𝐴 ↑_𝑜 𝑋))
40		onelon 5748	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_𝑜 𝑋) ∈ On ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) → 𝑍 ∈ On)
41	24, 39, 40	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ On)
42		oaword1 7632	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ⊆ (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍))
44	20	simprd 479	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍) = 𝐶)
45	43, 44	sseqtrd 3641	. . . . . . . 8 ⊢ (𝜑 → ((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) ⊆ 𝐶)
46	36, 45	sstrd 3613	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ⊆ 𝐶)
47		oecl 7617	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_𝑜 𝐵) ∈ On)
48	2, 3, 47	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝐵) ∈ On)
49		ontr2 5772	. . . . . . . 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 693	. . . . . . 7 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ⊆ 𝐶 ∧ 𝐶 ∈ (𝐴 ↑_𝑜 𝐵)) → (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵)))
51	46, 6, 50	mp2and 715	. . . . . 6 ⊢ (𝜑 → (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵))
52	9	simpld 475	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2_𝑜))
53		oeord 7668	. . . . . . 7 ⊢ ((𝑋 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ (On ∖ 2_𝑜)) → (𝑋 ∈ 𝐵 ↔ (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵)))
54	22, 3, 52, 53	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ (𝐴 ↑_𝑜 𝑋) ∈ (𝐴 ↑_𝑜 𝐵)))
55	51, 54	mpbird 247	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
56	2	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ On)
57	3	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐵 ∈ On)
58		suppssdm 7308	. . . . . . . . . . . . . . 15 ⊢ (𝐺 supp ∅) ⊆ dom 𝐺
59	1, 2, 3	cantnfs 8563	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
60	4, 59	mpbid 222	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
61	60	simpld 475	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
62		fdm 6051	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:𝐵⟶𝐴 → dom 𝐺 = 𝐵)
63	61, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom 𝐺 = 𝐵)
64	58, 63	syl5sseq 3653	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝐵)
65	64	sselda 3603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ 𝐵)
66		onelon 5748	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
67	57, 65, 66	syl2anc 693	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ On)
68		oecl 7617	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ↑_𝑜 𝑥) ∈ On)
69	56, 67, 68	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ∈ On)
70	61	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐺:𝐵⟶𝐴)
71	70, 65	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ∈ 𝐴)
72		onelon 5748	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑥) ∈ 𝐴) → (𝐺‘𝑥) ∈ On)
73	56, 71, 72	syl2anc 693	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ∈ On)
74		ffn 6045	. . . . . . . . . . . . . . 15 ⊢ (𝐺:𝐵⟶𝐴 → 𝐺 Fn 𝐵)
75	61, 74	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐺 Fn 𝐵)
76		0ex 4790	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ V
77	76	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∅ ∈ V)
78		elsuppfn 7303	. . . . . . . . . . . . . 14 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥 ∈ 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)))
79	75, 3, 77, 78	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥 ∈ (𝐺 supp ∅) ↔ (𝑥 ∈ 𝐵 ∧ (𝐺‘𝑥) ≠ ∅)))
80	79	simplbda 654	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐺‘𝑥) ≠ ∅)
81		on0eln0 5780	. . . . . . . . . . . . 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 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ∅ ∈ (𝐺‘𝑥))
84		omword1 7653	. . . . . . . . . . 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 1325	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ⊆ ((𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐺‘𝑥)))
86		eqid 2622	. . . . . . . . . . . 12 ⊢ OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
87	4	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐺 ∈ 𝑆)
88		eqid 2622	. . . . . . . . . . . 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 8568	. . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)
92	89, 91	sseqtrd 3641	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → ((𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐺‘𝑥)) ⊆ 𝑍)
93	85, 92	sstrd 3613	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ⊆ 𝑍)
94	39	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑍 ∈ (𝐴 ↑_𝑜 𝑋))
95	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑋) ∈ On)
96		ontr2 5772	. . . . . . . . . 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 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (((𝐴 ↑_𝑜 𝑥) ⊆ 𝑍 ∧ 𝑍 ∈ (𝐴 ↑_𝑜 𝑋)) → (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
98	93, 94, 97	mp2and 715	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋))
99	22	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑋 ∈ On)
100	52	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝐴 ∈ (On ∖ 2_𝑜))
101		oeord 7668	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑋 ∈ On ∧ 𝐴 ∈ (On ∖ 2_𝑜)) → (𝑥 ∈ 𝑋 ↔ (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
102	67, 99, 100, 101	syl3anc 1326	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → (𝑥 ∈ 𝑋 ↔ (𝐴 ↑_𝑜 𝑥) ∈ (𝐴 ↑_𝑜 𝑋)))
103	98, 102	mpbird 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 supp ∅)) → 𝑥 ∈ 𝑋)
104	103	ex 450	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐺 supp ∅) → 𝑥 ∈ 𝑋))
105	104	ssrdv 3609	. . . . 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 8578	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
108	107	simprd 479	. . 3 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
109	90	oveq2d 6666	. . 3 ⊢ (𝜑 → (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (((𝐴 ↑_𝑜 𝑋) ·_𝑜 𝑌) +_𝑜 𝑍))
110	108, 109, 44	3eqtrd 2660	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = 𝐶)
111	1, 2, 3	cantnff 8571	. . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_𝑜 𝐵))
112		ffn 6045	. . . 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 475	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
115		fnfvelrn 6356	. . 3 ⊢ (((𝐴 CNF 𝐵) Fn 𝑆 ∧ 𝐹 ∈ 𝑆) → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
116	113, 114, 115	syl2anc 693	. 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ran (𝐴 CNF 𝐵))
117	110, 116	eqeltrrd 2702	1 ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ⟨cop 4183 ∪ cuni 4436 ∩ cint 4475 class class class wbr 4653 {copab 4712 ↦ cmpt 4729 E cep 5028 dom cdm 5114 ran crn 5115 Oncon0 5723 ℩cio 5849 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167 supp csupp 7295 seq_𝜔cseqom 7542 1_𝑜c1o 7553 2_𝑜c2o 7554 +_𝑜 coa 7557 ·_𝑜 comu 7558 ↑_𝑜 coe 7559 finSupp cfsupp 8275 OrdIsocoi 8414 CNF ccnf 8558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seqom 7543 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-oexp 7566 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-oi 8415 df-cnf 8559

This theorem is referenced by: cantnflem4 8589

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