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Theorem cantnfp1 9649

Description: If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)

**Hypotheses**
Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
cantnfp1.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
cantnfp1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
cantnfp1.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
cantnfp1.s	⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
cantnfp1.f	⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))

**Assertion**
Ref	Expression
cantnfp1	⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺))))

Distinct variable groups: 𝑡,𝐵 𝑡,𝐴 𝑡,𝑆 𝑡,𝐺 𝜑,𝑡 𝑡,𝑌 𝑡,𝑋 Allowed substitution hint: 𝐹(𝑡)

Proof of Theorem cantnfp1 Dummy variables 𝑘 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cantnfp1.f	. . . . . 6 ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))
2		cantnfs.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ On)
3		cantnfp1.x	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		onelon 6386	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
5	2, 3, 4	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ On)
6		eloni 6371	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ On → Ord 𝑋)
7		ordirr 6379	. . . . . . . . . . . 12 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
8	5, 6, 7	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
9		fvex 6895	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑋) ∈ V
10		dif1o 8484	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑋) ∈ (V ∖ 1_o) ↔ ((𝐺‘𝑋) ∈ V ∧ (𝐺‘𝑋) ≠ ∅))
11	9, 10	mpbiran 721	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑋) ∈ (V ∖ 1_o) ↔ (𝐺‘𝑋) ≠ ∅)
12		cantnfp1.g	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
13		cantnfs.s	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑆 = dom (𝐴 CNF 𝐵)
14		cantnfs.a	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐴 ∈ On)
15	13, 14, 2	cantnfs 9634	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	12, 15	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 499	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18	17	ffnd 6707	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		0ex 5272	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
20	19	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ V)
21		elsuppfn 8165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)))
22	18, 2, 20, 21	syl3anc 1396	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)))
23	11	bicomi 227	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑋) ≠ ∅ ↔ (𝐺‘𝑋) ∈ (V ∖ 1_o))
24	23	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐺‘𝑋) ≠ ∅ ↔ (𝐺‘𝑋) ∈ (V ∖ 1_o)))
25	24	anbi2d 641	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1_o))))
26	22, 25	bitrd 282	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1_o))))
27		cantnfp1.s	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
28	27	sseld 3944	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋 ∈ 𝑋))
29	26, 28	sylbird 263	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1_o)) → 𝑋 ∈ 𝑋))
30	3, 29	mpand 707	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐺‘𝑋) ∈ (V ∖ 1_o) → 𝑋 ∈ 𝑋))
31	11, 30	biimtrrid 246	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺‘𝑋) ≠ ∅ → 𝑋 ∈ 𝑋))
32	31	necon1bd 2982	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑋 → (𝐺‘𝑋) = ∅))
33	8, 32	mpd 16	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝑋) = ∅)
34	33	ad3antrrr 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → (𝐺‘𝑋) = ∅)
35		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
36	35	fveq2d 6886	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → (𝐺‘𝑡) = (𝐺‘𝑋))
37		simpllr 787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
38	34, 36, 37	3eqtr4rd 2815	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺‘𝑡))
39		eqidd 2770	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺‘𝑡) = (𝐺‘𝑡))
40	38, 39	ifeqda 4529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)) = (𝐺‘𝑡))
41	40	mpteq2dva 5208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
42	1, 41	eqtrid 2816	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐹 = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
43	17	feqmptd 6950	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
44	43	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
45	42, 44	eqtr4d 2807	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐹 = 𝐺)
46	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ∈ 𝑆)
47	45, 46	eqeltrd 2869	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐹 ∈ 𝑆)
48		oecl 8521	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑_o 𝐵) ∈ On)
49	14, 2, 48	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_o 𝐵) ∈ On)
50	13, 14, 2	cantnff 9642	. . . . . . . 8 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑_o 𝐵))
51	50, 12	ffvelcdmd 7081	. . . . . . 7 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴 ↑_o 𝐵))
52		onelon 6386	. . . . . . 7 ⊢ (((𝐴 ↑_o 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴 ↑_o 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
53	49, 51, 52	syl2anc 595	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
54	53	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
55		oa0r 8522	. . . . 5 ⊢ (((𝐴 CNF 𝐵)‘𝐺) ∈ On → (∅ +_o ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
56	54, 55	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (∅ +_o ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
57		oveq2 7419	. . . . . 6 ⊢ (𝑌 = ∅ → ((𝐴 ↑_o 𝑋) ·_o 𝑌) = ((𝐴 ↑_o 𝑋) ·_o ∅))
58		oecl 8521	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑_o 𝑋) ∈ On)
59	14, 5, 58	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑_o 𝑋) ∈ On)
60		om0 8501	. . . . . . 7 ⊢ ((𝐴 ↑_o 𝑋) ∈ On → ((𝐴 ↑_o 𝑋) ·_o ∅) = ∅)
61	59, 60	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑_o 𝑋) ·_o ∅) = ∅)
62	57, 61	sylan9eqr 2826	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 ↑_o 𝑋) ·_o 𝑌) = ∅)
63	62	oveq1d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +_o ((𝐴 CNF 𝐵)‘𝐺)))
64	45	fveq2d 6886	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
65	56, 63, 64	3eqtr4rd 2815	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)))
66	47, 65	jca 520	. 2 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺))))
67	14	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐴 ∈ On)
68	2	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐵 ∈ On)
69	12	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈ 𝑆)
70	3	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑋 ∈ 𝐵)
71		cantnfp1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
72	71	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ 𝐴)
73	27	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
74	13, 67, 68, 69, 70, 72, 73, 1	cantnfp1lem1 9646	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐹 ∈ 𝑆)
75		onelon 6386	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
76	14, 71, 75	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ On)
77		on0eln0 6419	. . . . . 6 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
78	76, 77	syl 18	. . . . 5 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
79	78	biimpar 482	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∅ ∈ 𝑌)
80		eqid 2769	. . . 4 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
81		eqid 2769	. . . 4 ⊢ seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·_o (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +_o 𝑧)), ∅) = seq_ω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑_o (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·_o (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +_o 𝑧)), ∅)
82		eqid 2769	. . . 4 ⊢ OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
83		eqid 2769	. . . 4 ⊢ 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)), ∅)
84	13, 67, 68, 69, 70, 72, 73, 1, 79, 80, 81, 82, 83	cantnfp1lem3 9648	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺)))
85	74, 84	jca 520	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺))))
86	66, 85	pm2.61dane 3051	1 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑_o 𝑋) ·_o 𝑌) +_o ((𝐴 CNF 𝐵)‘𝐺))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 Vcvv 3463 ∖ cdif 3910 ⊆ wss 3913 ∅c0 4294 ifcif 4492 class class class wbr 5113 ↦ cmpt 5196 E cep 5561 dom cdm 5662 Ord word 6360 Oncon0 6361 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 supp csupp 8155 seq_ωcseqom 8433 1_oc1o 8445 +_o coa 8449 ·_o comu 8450 ↑_o coe 8451 finSupp cfsupp 9320 OrdIsocoi 9470 CNF ccnf 9629

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-seqom 8434 df-1o 8452 df-2o 8453 df-oadd 8456 df-omul 8457 df-oexp 8458 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-oi 9471 df-cnf 9630

This theorem is referenced by: cantnflem1d 9656 cantnflem1 9657 cantnflem3 9659 cantnfresb 43942

W3C validator