Theorem cantnfp1 8578

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 5748	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ On)
5	2, 3, 4	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ On)
6		eloni 5733	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ On → Ord 𝑋)
7		ordirr 5741	. . . . . . . . . . . 12 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
8	5, 6, 7	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
9		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (𝐺‘𝑋) ∈ V
10		dif1o 7580	. . . . . . . . . . . . . 14 ⊢ ((𝐺‘𝑋) ∈ (V ∖ 1𝑜) ↔ ((𝐺‘𝑋) ∈ V ∧ (𝐺‘𝑋) ≠ ∅))
11	9, 10	mpbiran 953	. . . . . . . . . . . . 13 ⊢ ((𝐺‘𝑋) ∈ (V ∖ 1𝑜) ↔ (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 8563	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)))
16	12, 15	mpbid 222	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))
17	16	simpld 475	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)
18		ffn 6045	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺:𝐵⟶𝐴 → 𝐺 Fn 𝐵)
19	17, 18	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐺 Fn 𝐵)
20		0ex 4790	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ V
21	20	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∅ ∈ V)
22		elsuppfn 7303	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)))
23	19, 2, 21, 22	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)))
24	11	bicomi 214	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺‘𝑋) ≠ ∅ ↔ (𝐺‘𝑋) ∈ (V ∖ 1𝑜))
25	24	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝐺‘𝑋) ≠ ∅ ↔ (𝐺‘𝑋) ∈ (V ∖ 1𝑜)))
26	25	anbi2d 740	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1𝑜))))
27	23, 26	bitrd 268	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1𝑜))))
28		cantnfp1.s	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)
29	28	sseld 3602	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) → 𝑋 ∈ 𝑋))
30	27, 29	sylbird 250	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ∈ (V ∖ 1𝑜)) → 𝑋 ∈ 𝑋))
31	3, 30	mpand 711	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐺‘𝑋) ∈ (V ∖ 1𝑜) → 𝑋 ∈ 𝑋))
32	11, 31	syl5bir 233	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐺‘𝑋) ≠ ∅ → 𝑋 ∈ 𝑋))
33	32	necon1bd 2812	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑋 ∈ 𝑋 → (𝐺‘𝑋) = ∅))
34	8, 33	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺‘𝑋) = ∅)
35	34	ad3antrrr 766	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → (𝐺‘𝑋) = ∅)
36		simpr 477	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → 𝑡 = 𝑋)
37	36	fveq2d 6195	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → (𝐺‘𝑡) = (𝐺‘𝑋))
38		simpllr 799	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = ∅)
39	35, 37, 38	3eqtr4rd 2667	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ 𝑡 = 𝑋) → 𝑌 = (𝐺‘𝑡))
40		eqidd 2623	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) ∧ ¬ 𝑡 = 𝑋) → (𝐺‘𝑡) = (𝐺‘𝑡))
41	39, 40	ifeqda 4121	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = ∅) ∧ 𝑡 ∈ 𝐵) → if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)) = (𝐺‘𝑡))
42	41	mpteq2dva 4744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡))) = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
43	1, 42	syl5eq 2668	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐹 = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
44	17	feqmptd 6249	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
45	44	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = (𝑡 ∈ 𝐵 ↦ (𝐺‘𝑡)))
46	43, 45	eqtr4d 2659	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐹 = 𝐺)
47	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ∈ 𝑆)
48	46, 47	eqeltrd 2701	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐹 ∈ 𝑆)
49		oecl 7617	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
50	14, 2, 49	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) ∈ On)
51	13, 14, 2	cantnff 8571	. . . . . . . 8 ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑𝑜 𝐵))
52	51, 12	ffvelrnd 6360	. . . . . . 7 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴 ↑𝑜 𝐵))
53		onelon 5748	. . . . . . 7 ⊢ (((𝐴 ↑𝑜 𝐵) ∈ On ∧ ((𝐴 CNF 𝐵)‘𝐺) ∈ (𝐴 ↑𝑜 𝐵)) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
54	50, 52, 53	syl2anc 693	. . . . . 6 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
55	54	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐺) ∈ On)
56		oa0r 7618	. . . . 5 ⊢ (((𝐴 CNF 𝐵)‘𝐺) ∈ On → (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
57	55, 56	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = ((𝐴 CNF 𝐵)‘𝐺))
58		oveq2 6658	. . . . . 6 ⊢ (𝑌 = ∅ → ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) = ((𝐴 ↑𝑜 𝑋) ·𝑜 ∅))
59		oecl 7617	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑋 ∈ On) → (𝐴 ↑𝑜 𝑋) ∈ On)
60	14, 5, 59	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐴 ↑𝑜 𝑋) ∈ On)
61		om0 7597	. . . . . . 7 ⊢ ((𝐴 ↑𝑜 𝑋) ∈ On → ((𝐴 ↑𝑜 𝑋) ·𝑜 ∅) = ∅)
62	60, 61	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝑋) ·𝑜 ∅) = ∅)
63	58, 62	sylan9eqr 2678	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) = ∅)
64	63	oveq1d 6665	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)) = (∅ +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
65	46	fveq2d 6195	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = ((𝐴 CNF 𝐵)‘𝐺))
66	57, 64, 65	3eqtr4rd 2667	. . 3 ⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
67	48, 66	jca 554	. 2 ⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
68	14	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐴 ∈ On)
69	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐵 ∈ On)
70	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈ 𝑆)
71	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑋 ∈ 𝐵)
72		cantnfp1.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
73	72	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ 𝐴)
74	28	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 supp ∅) ⊆ 𝑋)
75	13, 68, 69, 70, 71, 73, 74, 1	cantnfp1lem1 8575	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐹 ∈ 𝑆)
76		onelon 5748	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ On)
77	14, 72, 76	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ On)
78		on0eln0 5780	. . . . . 6 ⊢ (𝑌 ∈ On → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
79	77, 78	syl 17	. . . . 5 ⊢ (𝜑 → (∅ ∈ 𝑌 ↔ 𝑌 ≠ ∅))
80	79	biimpar 502	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∅ ∈ 𝑌)
81		eqid 2622	. . . 4 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅))
82		eqid 2622	. . . 4 ⊢ seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·𝑜 (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·𝑜 (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
83		eqid 2622	. . . 4 ⊢ OrdIso( E , (𝐺 supp ∅)) = OrdIso( E , (𝐺 supp ∅))
84		eqid 2622	. . . 4 ⊢ seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑𝑜 (OrdIso( E , (𝐺 supp ∅))‘𝑘)) ·𝑜 (𝐺‘(OrdIso( E , (𝐺 supp ∅))‘𝑘))) +𝑜 𝑧)), ∅)
85	13, 68, 69, 70, 71, 73, 74, 1, 80, 81, 82, 83, 84	cantnfp1lem3 8577	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺)))
86	75, 85	jca 554	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))
87	67, 86	pm2.61dane 2881	1 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑𝑜 𝑋) ·𝑜 𝑌) +𝑜 ((𝐴 CNF 𝐵)‘𝐺))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729   E cep 5028  dom cdm 5114  Ord word 5722  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   supp csupp 7295  seq_𝜔cseqom 7542  1_𝑜c1o 7553   +_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:  cantnflem1d  8585  cantnflem1  8586  cantnflem3  8588