Theorem cantnfle 8568

Description: A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑_𝑜 𝑥) ·_𝑜 (𝐹‘𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶 ∈ 𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfcl.g	⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
cantnfcl.f	⊢ (𝜑 → 𝐹 ∈ 𝑆)
cantnfval.h	⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)
cantnfle.c	⊢ (𝜑 → 𝐶 ∈ 𝐵)

Assertion

Ref	Expression
cantnfle	⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))

Distinct variable groups:   𝑧,𝑘,𝐵   𝑧,𝐶   𝐴,𝑘,𝑧   𝑘,𝐹,𝑧   𝑆,𝑘,𝑧   𝑘,𝐺,𝑧   𝜑,𝑘,𝑧

Allowed substitution hints:   𝐶(𝑘)   𝐻(𝑧,𝑘)

Proof of Theorem cantnfle

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 ⊢ ((𝐹‘𝐶) = ∅ → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) = ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅))
2	1	sseq1d 3632	. 2 ⊢ ((𝐹‘𝐶) = ∅ → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) ⊆ ((𝐴 CNF 𝐵)‘𝐹)))
3		cantnfs.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ On)
4		suppssdm 7308	. . . . . . . . . . 11 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹
5		cantnfcl.f	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ 𝑆)
6		cantnfs.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
7		cantnfs.a	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ On)
8	6, 7, 3	cantnfs 8563	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
9	5, 8	mpbid 222	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
10	9	simpld 475	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		fdm 6051	. . . . . . . . . . . 12 ⊢ (𝐹:𝐵⟶𝐴 → dom 𝐹 = 𝐵)
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐵)
13	4, 12	syl5sseq 3653	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵)
14	3, 13	ssexd 4805	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
15		cantnfcl.g	. . . . . . . . . . 11 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
16	6, 7, 3, 15, 5	cantnfcl 8564	. . . . . . . . . 10 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
17	16	simpld 475	. . . . . . . . 9 ⊢ (𝜑 → E We (𝐹 supp ∅))
18	15	oiiso 8442	. . . . . . . . 9 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
19	14, 17, 18	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
20		isof1o 6573	. . . . . . . 8 ⊢ (𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) → 𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅))
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅))
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅))
23		f1ocnv 6149	. . . . . 6 ⊢ (𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅) → ◡𝐺:(𝐹 supp ∅)–1-1-onto→dom 𝐺)
24		f1of 6137	. . . . . 6 ⊢ (◡𝐺:(𝐹 supp ∅)–1-1-onto→dom 𝐺 → ◡𝐺:(𝐹 supp ∅)⟶dom 𝐺)
25	22, 23, 24	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ◡𝐺:(𝐹 supp ∅)⟶dom 𝐺)
26		cantnfle.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
27	26	anim1i 592	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (𝐶 ∈ 𝐵 ∧ (𝐹‘𝐶) ≠ ∅))
28	10	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐹:𝐵⟶𝐴)
29		ffn 6045	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐴 → 𝐹 Fn 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐹 Fn 𝐵)
31	3	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐵 ∈ On)
32		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
33	32	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ∅ ∈ V)
34		elsuppfn 7303	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝐶 ∈ (𝐹 supp ∅) ↔ (𝐶 ∈ 𝐵 ∧ (𝐹‘𝐶) ≠ ∅)))
35	30, 31, 33, 34	syl3anc 1326	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (𝐶 ∈ (𝐹 supp ∅) ↔ (𝐶 ∈ 𝐵 ∧ (𝐹‘𝐶) ≠ ∅)))
36	27, 35	mpbird 247	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐶 ∈ (𝐹 supp ∅))
37	25, 36	ffvelrnd 6360	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (◡𝐺‘𝐶) ∈ dom 𝐺)
38	16	simprd 479	. . . . . 6 ⊢ (𝜑 → dom 𝐺 ∈ ω)
39	38	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → dom 𝐺 ∈ ω)
40		eqimss 3657	. . . . . . . . . 10 ⊢ (𝑥 = dom 𝐺 → 𝑥 ⊆ dom 𝐺)
41	40	biantrurd 529	. . . . . . . . 9 ⊢ (𝑥 = dom 𝐺 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥)))
42		eleq2 2690	. . . . . . . . 9 ⊢ (𝑥 = dom 𝐺 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ dom 𝐺))
43	41, 42	bitr3d 270	. . . . . . . 8 ⊢ (𝑥 = dom 𝐺 → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (◡𝐺‘𝐶) ∈ dom 𝐺))
44		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = dom 𝐺 → (𝐻‘𝑥) = (𝐻‘dom 𝐺))
45	44	sseq2d 3633	. . . . . . . 8 ⊢ (𝑥 = dom 𝐺 → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺)))
46	43, 45	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = dom 𝐺 → (((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥)) ↔ ((◡𝐺‘𝐶) ∈ dom 𝐺 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺))))
47	46	imbi2d 330	. . . . . 6 ⊢ (𝑥 = dom 𝐺 → (((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥))) ↔ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((◡𝐺‘𝐶) ∈ dom 𝐺 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺)))))
48		sseq1 3626	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
49		eleq2 2690	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ ∅))
50	48, 49	anbi12d 747	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (∅ ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ ∅)))
51		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐻‘𝑥) = (𝐻‘∅))
52	51	sseq2d 3633	. . . . . . . 8 ⊢ (𝑥 = ∅ → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅)))
53	50, 52	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = ∅ → (((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥)) ↔ ((∅ ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ ∅) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅))))
54		sseq1 3626	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ dom 𝐺 ↔ 𝑦 ⊆ dom 𝐺))
55		eleq2 2690	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ 𝑦))
56	54, 55	anbi12d 747	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)))
57		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐻‘𝑥) = (𝐻‘𝑦))
58	57	sseq2d 3633	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)))
59	56, 58	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥)) ↔ ((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦))))
60		sseq1 3626	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ dom 𝐺 ↔ suc 𝑦 ⊆ dom 𝐺))
61		eleq2 2690	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ suc 𝑦))
62	60, 61	anbi12d 747	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ suc 𝑦)))
63		fveq2 6191	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐻‘𝑥) = (𝐻‘suc 𝑦))
64	63	sseq2d 3633	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))
65	62, 64	imbi12d 334	. . . . . . 7 ⊢ (𝑥 = suc 𝑦 → (((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥)) ↔ ((suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ suc 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
66		noel 3919	. . . . . . . . . 10 ⊢ ¬ (◡𝐺‘𝐶) ∈ ∅
67	66	pm2.21i 116	. . . . . . . . 9 ⊢ ((◡𝐺‘𝐶) ∈ ∅ → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅))
68	67	adantl 482	. . . . . . . 8 ⊢ ((∅ ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ ∅) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅))
69	68	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((∅ ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ ∅) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅)))
70		fvex 6201	. . . . . . . . . . . 12 ⊢ (◡𝐺‘𝐶) ∈ V
71	70	elsuc 5794	. . . . . . . . . . 11 ⊢ ((◡𝐺‘𝐶) ∈ suc 𝑦 ↔ ((◡𝐺‘𝐶) ∈ 𝑦 ∨ (◡𝐺‘𝐶) = 𝑦))
72		sssucid 5802	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ⊆ suc 𝑦
73		sstr 3611	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ⊆ dom 𝐺)
74	72, 73	mpan 706	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑦 ⊆ dom 𝐺 → 𝑦 ⊆ dom 𝐺)
75	74	ad2antrl 764	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → 𝑦 ⊆ dom 𝐺)
76		simprr 796	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → (◡𝐺‘𝐶) ∈ 𝑦)
77		pm2.27 42	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)))
78	75, 76, 77	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)))
79		cantnfval.h	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (𝐺‘𝑘)) ·𝑜 (𝐹‘(𝐺‘𝑘))) +𝑜 𝑧)), ∅)
80	79	cantnfvalf 8562	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐻:ω⟶On
81	80	ffvelrni 6358	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → (𝐻‘𝑦) ∈ On)
82	81	ad2antlr 763	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐻‘𝑦) ∈ On)
83	7	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝐴 ∈ On)
84	3	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝐵 ∈ On)
85	13	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐹 supp ∅) ⊆ 𝐵)
86		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → suc 𝑦 ⊆ dom 𝐺)
87		sucidg 5803	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ω → 𝑦 ∈ suc 𝑦)
88	87	ad2antlr 763	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ∈ suc 𝑦)
89	86, 88	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ∈ dom 𝐺)
90	15	oif 8435	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
91	90	ffvelrni 6358	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ dom 𝐺 → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
92	89, 91	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐺‘𝑦) ∈ (𝐹 supp ∅))
93	85, 92	sseldd 3604	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐺‘𝑦) ∈ 𝐵)
94		onelon 5748	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ On ∧ (𝐺‘𝑦) ∈ 𝐵) → (𝐺‘𝑦) ∈ On)
95	84, 93, 94	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐺‘𝑦) ∈ On)
96		oecl 7617	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑦) ∈ On) → (𝐴 ↑𝑜 (𝐺‘𝑦)) ∈ On)
97	83, 95, 96	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐴 ↑𝑜 (𝐺‘𝑦)) ∈ On)
98	10	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝐹:𝐵⟶𝐴)
99	98, 93	ffvelrnd 6360	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐹‘(𝐺‘𝑦)) ∈ 𝐴)
100		onelon 5748	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ (𝐹‘(𝐺‘𝑦)) ∈ 𝐴) → (𝐹‘(𝐺‘𝑦)) ∈ On)
101	83, 99, 100	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐹‘(𝐺‘𝑦)) ∈ On)
102		omcl 7616	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ∈ On ∧ (𝐹‘(𝐺‘𝑦)) ∈ On) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ∈ On)
103	97, 101, 102	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ∈ On)
104		oaword2 7633	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐻‘𝑦) ∈ On ∧ ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ∈ On) → (𝐻‘𝑦) ⊆ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
105	82, 103, 104	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐻‘𝑦) ⊆ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
106		simplll 798	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝜑)
107		simplr 792	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ∈ ω)
108	6, 7, 3, 15, 5, 79	cantnfsuc 8567	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (𝐻‘suc 𝑦) = (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
109	106, 107, 108	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐻‘suc 𝑦) = (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
110	105, 109	sseqtr4d 3642	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐻‘𝑦) ⊆ (𝐻‘suc 𝑦))
111		sstr 3611	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦) ∧ (𝐻‘𝑦) ⊆ (𝐻‘suc 𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))
112	111	expcom 451	. . . . . . . . . . . . . . . 16 ⊢ ((𝐻‘𝑦) ⊆ (𝐻‘suc 𝑦) → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))
113	110, 112	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))
114	113	adantrr 753	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))
115	78, 114	syld 47	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))
116	115	expr 643	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((◡𝐺‘𝐶) ∈ 𝑦 → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
117		simprr 796	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (◡𝐺‘𝐶) = 𝑦)
118	117	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐺‘(◡𝐺‘𝐶)) = (𝐺‘𝑦))
119		f1ocnvfv2 6533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅) ∧ 𝐶 ∈ (𝐹 supp ∅)) → (𝐺‘(◡𝐺‘𝐶)) = 𝐶)
120	22, 36, 119	syl2anc 693	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (𝐺‘(◡𝐺‘𝐶)) = 𝐶)
121	120	ad2antrr 762	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐺‘(◡𝐺‘𝐶)) = 𝐶)
122	118, 121	eqtr3d 2658	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐺‘𝑦) = 𝐶)
123	122	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐴 ↑𝑜 (𝐺‘𝑦)) = (𝐴 ↑𝑜 𝐶))
124	122	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐹‘(𝐺‘𝑦)) = (𝐹‘𝐶))
125	123, 124	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) = ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)))
126		oaword1 7632	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ∈ On ∧ (𝐻‘𝑦) ∈ On) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ⊆ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
127	103, 82, 126	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ⊆ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
128	127	adantrr 753	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ⊆ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
129	125, 128	eqsstr3d 3640	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
130	109	adantrr 753	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐻‘suc 𝑦) = (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
131	129, 130	sseqtr4d 3642	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))
132	131	expr 643	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((◡𝐺‘𝐶) = 𝑦 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))
133	132	a1dd 50	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((◡𝐺‘𝐶) = 𝑦 → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
134	116, 133	jaod 395	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (((◡𝐺‘𝐶) ∈ 𝑦 ∨ (◡𝐺‘𝐶) = 𝑦) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
135	71, 134	syl5bi 232	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((◡𝐺‘𝐶) ∈ suc 𝑦 → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
136	135	expimpd 629	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) → ((suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ suc 𝑦) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
137	136	com23 86	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ suc 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦))))
138	137	expcom 451	. . . . . . 7 ⊢ (𝑦 ∈ ω → ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (((𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)) → ((suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ suc 𝑦) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘suc 𝑦)))))
139	53, 59, 65, 69, 138	finds2 7094	. . . . . 6 ⊢ (𝑥 ∈ ω → ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥))))
140	47, 139	vtoclga 3272	. . . . 5 ⊢ (dom 𝐺 ∈ ω → ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((◡𝐺‘𝐶) ∈ dom 𝐺 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺))))
141	39, 140	mpcom 38	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((◡𝐺‘𝐶) ∈ dom 𝐺 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺)))
142	37, 141	mpd 15	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺))
143	6, 7, 3, 15, 5, 79	cantnfval 8565	. . . 4 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
144	143	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
145	142, 144	sseqtr4d 3642	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
146		onelon 5748	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ On)
147	3, 26, 146	syl2anc 693	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ On)
148		oecl 7617	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
149	7, 147, 148	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐶) ∈ On)
150		om0 7597	. . . 4 ⊢ ((𝐴 ↑𝑜 𝐶) ∈ On → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) = ∅)
151	149, 150	syl 17	. . 3 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) = ∅)
152		0ss 3972	. . 3 ⊢ ∅ ⊆ ((𝐴 CNF 𝐵)‘𝐹)
153	151, 152	syl6eqss 3655	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
154	2, 145, 153	pm2.61ne 2879	1 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   class class class wbr 4653   E cep 5028   We wwe 5072  ^◡ccnv 5113  dom cdm 5114  Oncon0 5723  suc csuc 5725   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065   supp csupp 7295  seq_𝜔cseqom 7542   +_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-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-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:  cantnflem3  8588