Theorem cantnfle 8451

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 6557	. . 3 ⊢ ((𝐹‘𝐶) = ∅ → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) = ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅))
2	1	sseq1d 3595	. 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 7195	. . . . . . . . . . 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 8446	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
9	5, 8	mpbid 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))
10	9	simpld 474	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴)
11		fdm 5964	. . . . . . . . . . . 12 ⊢ (𝐹:𝐵⟶𝐴 → dom 𝐹 = 𝐵)
12	10, 11	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝐵)
13	4, 12	syl5sseq 3616	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵)
14	3, 13	ssexd 4733	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V)
15		cantnfcl.g	. . . . . . . . . . 11 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))
16	6, 7, 3, 15, 5	cantnfcl 8447	. . . . . . . . . 10 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
17	16	simpld 474	. . . . . . . . 9 ⊢ (𝜑 → E We (𝐹 supp ∅))
18	15	oiiso 8325	. . . . . . . . 9 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
19	14, 17, 18	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
20		isof1o 6473	. . . . . . . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅))
23		f1ocnv 6062	. . . . . 6 ⊢ (𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅) → ◡𝐺:(𝐹 supp ∅)–1-1-onto→dom 𝐺)
24		f1of 6050	. . . . . 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 590	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (𝐶 ∈ 𝐵 ∧ (𝐹‘𝐶) ≠ ∅))
28	10	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐹:𝐵⟶𝐴)
29		ffn 5958	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐴 → 𝐹 Fn 𝐵)
30	28, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐹 Fn 𝐵)
31	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐵 ∈ On)
32		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
33	32	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ∅ ∈ V)
34		elsuppfn 7190	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝐶 ∈ (𝐹 supp ∅) ↔ (𝐶 ∈ 𝐵 ∧ (𝐹‘𝐶) ≠ ∅)))
35	30, 31, 33, 34	syl3anc 1318	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (𝐶 ∈ (𝐹 supp ∅) ↔ (𝐶 ∈ 𝐵 ∧ (𝐹‘𝐶) ≠ ∅)))
36	27, 35	mpbird 246	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → 𝐶 ∈ (𝐹 supp ∅))
37	25, 36	ffvelrnd 6268	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (◡𝐺‘𝐶) ∈ dom 𝐺)
38	16	simprd 478	. . . . . 6 ⊢ (𝜑 → dom 𝐺 ∈ ω)
39	38	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → dom 𝐺 ∈ ω)
40		eqimss 3620	. . . . . . . . . 10 ⊢ (𝑥 = dom 𝐺 → 𝑥 ⊆ dom 𝐺)
41	40	biantrurd 528	. . . . . . . . 9 ⊢ (𝑥 = dom 𝐺 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥)))
42		eleq2 2677	. . . . . . . . 9 ⊢ (𝑥 = dom 𝐺 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ dom 𝐺))
43	41, 42	bitr3d 269	. . . . . . . 8 ⊢ (𝑥 = dom 𝐺 → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (◡𝐺‘𝐶) ∈ dom 𝐺))
44		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = dom 𝐺 → (𝐻‘𝑥) = (𝐻‘dom 𝐺))
45	44	sseq2d 3596	. . . . . . . 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 333	. . . . . . 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 329	. . . . . 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 3589	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ⊆ dom 𝐺 ↔ ∅ ⊆ dom 𝐺))
49		eleq2 2677	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ ∅))
50	48, 49	anbi12d 743	. . . . . . . 8 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (∅ ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ ∅)))
51		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝐻‘𝑥) = (𝐻‘∅))
52	51	sseq2d 3596	. . . . . . . 8 ⊢ (𝑥 = ∅ → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅)))
53	50, 52	imbi12d 333	. . . . . . 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 3589	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ dom 𝐺 ↔ 𝑦 ⊆ dom 𝐺))
55		eleq2 2677	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ 𝑦))
56	54, 55	anbi12d 743	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)))
57		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐻‘𝑥) = (𝐻‘𝑦))
58	57	sseq2d 3596	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥) ↔ ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑦)))
59	56, 58	imbi12d 333	. . . . . . 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 3589	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝑥 ⊆ dom 𝐺 ↔ suc 𝑦 ⊆ dom 𝐺))
61		eleq2 2677	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → ((◡𝐺‘𝐶) ∈ 𝑥 ↔ (◡𝐺‘𝐶) ∈ suc 𝑦))
62	60, 61	anbi12d 743	. . . . . . . 8 ⊢ (𝑥 = suc 𝑦 → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) ↔ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ suc 𝑦)))
63		fveq2 6103	. . . . . . . . 9 ⊢ (𝑥 = suc 𝑦 → (𝐻‘𝑥) = (𝐻‘suc 𝑦))
64	63	sseq2d 3596	. . . . . . . 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 333	. . . . . . 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 3878	. . . . . . . . . 10 ⊢ ¬ (◡𝐺‘𝐶) ∈ ∅
67	66	pm2.21i 115	. . . . . . . . 9 ⊢ ((◡𝐺‘𝐶) ∈ ∅ → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘∅))
68	67	adantl 481	. . . . . . . 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 6113	. . . . . . . . . . . 12 ⊢ (◡𝐺‘𝐶) ∈ V
71	70	elsuc 5711	. . . . . . . . . . 11 ⊢ ((◡𝐺‘𝐶) ∈ suc 𝑦 ↔ ((◡𝐺‘𝐶) ∈ 𝑦 ∨ (◡𝐺‘𝐶) = 𝑦))
72		sssucid 5719	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ⊆ suc 𝑦
73		sstr 3576	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ⊆ suc 𝑦 ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ⊆ dom 𝐺)
74	72, 73	mpan 702	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑦 ⊆ dom 𝐺 → 𝑦 ⊆ dom 𝐺)
75	74	ad2antrl 760	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → 𝑦 ⊆ dom 𝐺)
76		simprr 792	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑦)) → (◡𝐺‘𝐶) ∈ 𝑦)
77		pm2.27 41	. . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . 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 8445	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐻:ω⟶On
81	80	ffvelrni 6266	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ω → (𝐻‘𝑦) ∈ On)
82	81	ad2antlr 759	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐻‘𝑦) ∈ On)
83	7	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝐴 ∈ On)
84	3	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝐵 ∈ On)
85	13	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐹 supp ∅) ⊆ 𝐵)
86		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → suc 𝑦 ⊆ dom 𝐺)
87		sucidg 5720	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ω → 𝑦 ∈ suc 𝑦)
88	87	ad2antlr 759	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ∈ suc 𝑦)
89	86, 88	sseldd 3569	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ∈ dom 𝐺)
90	15	oif 8318	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝐺:dom 𝐺⟶(𝐹 supp ∅)
91	90	ffvelrni 6266	. . . . . . . . . . . . . . . . . . . . . . 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 3569	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐺‘𝑦) ∈ 𝐵)
94		onelon 5665	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐵 ∈ On ∧ (𝐺‘𝑦) ∈ 𝐵) → (𝐺‘𝑦) ∈ On)
95	84, 93, 94	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐺‘𝑦) ∈ On)
96		oecl 7504	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ (𝐺‘𝑦) ∈ On) → (𝐴 ↑𝑜 (𝐺‘𝑦)) ∈ On)
97	83, 95, 96	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐴 ↑𝑜 (𝐺‘𝑦)) ∈ On)
98	10	ad3antrrr 762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝐹:𝐵⟶𝐴)
99	98, 93	ffvelrnd 6268	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐹‘(𝐺‘𝑦)) ∈ 𝐴)
100		onelon 5665	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ (𝐹‘(𝐺‘𝑦)) ∈ 𝐴) → (𝐹‘(𝐺‘𝑦)) ∈ On)
101	83, 99, 100	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐹‘(𝐺‘𝑦)) ∈ On)
102		omcl 7503	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ↑𝑜 (𝐺‘𝑦)) ∈ On ∧ (𝐹‘(𝐺‘𝑦)) ∈ On) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ∈ On)
103	97, 101, 102	syl2anc 691	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → ((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) ∈ On)
104		oaword2 7520	. . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝜑)
107		simplr 788	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → 𝑦 ∈ ω)
108	6, 7, 3, 15, 5, 79	cantnfsuc 8450	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → (𝐻‘suc 𝑦) = (((𝐴 ↑𝑜 (𝐺‘𝑦)) ·𝑜 (𝐹‘(𝐺‘𝑦))) +𝑜 (𝐻‘𝑦)))
109	106, 107, 108	syl2anc 691	. . . . . . . . . . . . . . . . 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 3605	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ suc 𝑦 ⊆ dom 𝐺) → (𝐻‘𝑦) ⊆ (𝐻‘suc 𝑦))
111		sstr 3576	. . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . 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 46	. . . . . . . . . . . . 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 641	. . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (◡𝐺‘𝐶) = 𝑦)
118	117	fveq2d 6107	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐺‘(◡𝐺‘𝐶)) = (𝐺‘𝑦))
119		f1ocnvfv2 6433	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐺:dom 𝐺–1-1-onto→(𝐹 supp ∅) ∧ 𝐶 ∈ (𝐹 supp ∅)) → (𝐺‘(◡𝐺‘𝐶)) = 𝐶)
120	22, 36, 119	syl2anc 691	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → (𝐺‘(◡𝐺‘𝐶)) = 𝐶)
121	120	ad2antrr 758	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐺‘(◡𝐺‘𝐶)) = 𝐶)
122	118, 121	eqtr3d 2646	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐺‘𝑦) = 𝐶)
123	122	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐴 ↑𝑜 (𝐺‘𝑦)) = (𝐴 ↑𝑜 𝐶))
124	122	fveq2d 6107	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) ∧ 𝑦 ∈ ω) ∧ (suc 𝑦 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) = 𝑦)) → (𝐹‘(𝐺‘𝑦)) = (𝐹‘𝐶))
125	123, 124	oveq12d 6567	. . . . . . . . . . . . . . . 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 7519	. . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . . 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 3603	. . . . . . . . . . . . . . 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 749	. . . . . . . . . . . . . . 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 3605	. . . . . . . . . . . . . 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 641	. . . . . . . . . . . . 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 48	. . . . . . . . . . . 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 394	. . . . . . . . . . 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 231	. . . . . . . . . 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 627	. . . . . . . . 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 84	. . . . . . . 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 450	. . . . . . 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 6986	. . . . . 6 ⊢ (𝑥 ∈ ω → ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝑥 ⊆ dom 𝐺 ∧ (◡𝐺‘𝐶) ∈ 𝑥) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘𝑥))))
140	47, 139	vtoclga 3245	. . . . 5 ⊢ (dom 𝐺 ∈ ω → ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((◡𝐺‘𝐶) ∈ dom 𝐺 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ (𝐻‘dom 𝐺))))
141	39, 140	mpcom 37	. . . 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 8448	. . . 4 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
144	143	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
145	142, 144	sseqtr4d 3605	. 2 ⊢ ((𝜑 ∧ (𝐹‘𝐶) ≠ ∅) → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
146		onelon 5665	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ 𝐵) → 𝐶 ∈ On)
147	3, 26, 146	syl2anc 691	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ On)
148		oecl 7504	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑𝑜 𝐶) ∈ On)
149	7, 147, 148	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐶) ∈ On)
150		om0 7484	. . . 4 ⊢ ((𝐴 ↑𝑜 𝐶) ∈ On → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) = ∅)
151	149, 150	syl 17	. . 3 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) = ∅)
152		0ss 3924	. . 3 ⊢ ∅ ⊆ ((𝐴 CNF 𝐵)‘𝐹)
153	151, 152	syl6eqss 3618	. 2 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 ∅) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
154	2, 145, 153	pm2.61ne 2867	1 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐶) ·𝑜 (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))

Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   E cep 4947   We wwe 4996  ^◡ccnv 5037  dom cdm 5038  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq_𝜔cseqom 7429   +_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-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-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:  cantnflem3  8471