Theorem hashdom 13029

Description: Dominance relation for the size function. (Contributed by Mario Carneiro, 22-Sep-2013.) (Revised by Mario Carneiro, 22-Apr-2015.)

Assertion

Ref	Expression
hashdom	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵))

Proof of Theorem hashdom

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

Step	Hyp	Ref	Expression
1		fzfi 12633	. . . . . . . 8 ⊢ (1...((#‘𝐵) − (#‘𝐴))) ∈ Fin
2		ficardom 8670	. . . . . . . 8 ⊢ ((1...((#‘𝐵) − (#‘𝐴))) ∈ Fin → (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω)
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω
4		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
5	4	hashgval 12982	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (#‘𝐴))
6	5	ad2antrr 758	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (#‘𝐴))
7	4	hashgval 12982	. . . . . . . . . . . . . 14 ⊢ ((1...((#‘𝐵) − (#‘𝐴))) ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = (#‘(1...((#‘𝐵) − (#‘𝐴)))))
8	1, 7	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = (#‘(1...((#‘𝐵) − (#‘𝐴))))
9		hashcl 13009	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ0)
10	9	ad2antrr 758	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ∈ ℕ0)
11		hashcl 13009	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℕ0)
12	11	ad2antlr 759	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐵) ∈ ℕ0)
13		simpr 476	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵))
14		nn0sub2 11315	. . . . . . . . . . . . . . 15 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ (#‘𝐵) ∈ ℕ0 ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐵) − (#‘𝐴)) ∈ ℕ0)
15	10, 12, 13, 14	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐵) − (#‘𝐴)) ∈ ℕ0)
16		hashfz1 12996	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐵) − (#‘𝐴)) ∈ ℕ0 → (#‘(1...((#‘𝐵) − (#‘𝐴)))) = ((#‘𝐵) − (#‘𝐴)))
17	15, 16	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (#‘(1...((#‘𝐵) − (#‘𝐴)))) = ((#‘𝐵) − (#‘𝐴)))
18	8, 17	syl5eq 2656	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴))))) = ((#‘𝐵) − (#‘𝐴)))
19	6, 18	oveq12d 6567	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴)))))) = ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))))
20	9	nn0cnd 11230	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℂ)
21	11	nn0cnd 11230	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (#‘𝐵) ∈ ℂ)
22		pncan3 10168	. . . . . . . . . . . . 13 ⊢ (((#‘𝐴) ∈ ℂ ∧ (#‘𝐵) ∈ ℂ) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
23	20, 21, 22	syl2an 493	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
24	23	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((#‘𝐴) + ((#‘𝐵) − (#‘𝐴))) = (#‘𝐵))
25	19, 24	eqtrd 2644	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴)))))) = (#‘𝐵))
26		ficardom 8670	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
27	26	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (card‘𝐴) ∈ ω)
28	4	hashgadd 13027	. . . . . . . . . . 11 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴)))))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴)))))))
29	27, 3, 28	sylancl 693	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴)))))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((#‘𝐵) − (#‘𝐴)))))))
30	4	hashgval 12982	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
31	30	ad2antlr 759	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
32	25, 29, 31	3eqtr4d 2654	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴)))))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))
33	32	fveq2d 6107	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))) = (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))))
34	4	hashgf1o 12632	. . . . . . . . 9 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0
35		nnacl 7578	. . . . . . . . . 10 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω)
36	27, 3, 35	sylancl 693	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω)
37		f1ocnvfv1 6432	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 ∧ ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) ∈ ω) → (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))) = ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))
38	34, 36, 37	sylancr 694	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))) = ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))
39		ficardom 8670	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
40	39	ad2antlr 759	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (card‘𝐵) ∈ ω)
41		f1ocnvfv1 6432	. . . . . . . . 9 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ0 ∧ (card‘𝐵) ∈ ω) → (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = (card‘𝐵))
42	34, 40, 41	sylancr 694	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → (◡(rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵))) = (card‘𝐵))
43	33, 38, 42	3eqtr3d 2652	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵))
44		oveq2 6557	. . . . . . . . 9 ⊢ (𝑦 = (card‘(1...((#‘𝐵) − (#‘𝐴)))) → ((card‘𝐴) +𝑜 𝑦) = ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))))
45	44	eqeq1d 2612	. . . . . . . 8 ⊢ (𝑦 = (card‘(1...((#‘𝐵) − (#‘𝐴)))) → (((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵)))
46	45	rspcev 3282	. . . . . . 7 ⊢ (((card‘(1...((#‘𝐵) − (#‘𝐴)))) ∈ ω ∧ ((card‘𝐴) +𝑜 (card‘(1...((#‘𝐵) − (#‘𝐴))))) = (card‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵))
47	3, 43, 46	sylancr 694	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (#‘𝐴) ≤ (#‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵))
48	47	ex 449	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
49		cardnn 8672	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
50	49	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
51	50	oveq2d 6565	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +𝑜 (card‘𝑦)) = ((card‘𝐴) +𝑜 𝑦))
52	51	eqeq1d 2612	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
53		fveq2 6103	. . . . . . . 8 ⊢ (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)))
54		nnfi 8038	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ Fin)
55		ficardom 8670	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
56	4	hashgadd 13027	. . . . . . . . . . . . . 14 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝑦) ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))))
57	26, 55, 56	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))))
58	4	hashgval 12982	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (#‘𝑦))
59	5, 58	oveqan12d 6568	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) + ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
60	57, 59	eqtrd 2644	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
61	60	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((#‘𝐴) + (#‘𝑦)))
62	30	ad2antlr 759	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (#‘𝐵))
63	61, 62	eqeq12d 2625	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) ↔ ((#‘𝐴) + (#‘𝑦)) = (#‘𝐵)))
64		hashcl 13009	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (#‘𝑦) ∈ ℕ0)
65	64	nn0ge0d 11231	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → 0 ≤ (#‘𝑦))
66	65	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (#‘𝑦))
67	9	nn0red 11229	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ)
68	64	nn0red 11229	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (#‘𝑦) ∈ ℝ)
69		addge01 10417	. . . . . . . . . . . . . 14 ⊢ (((#‘𝐴) ∈ ℝ ∧ (#‘𝑦) ∈ ℝ) → (0 ≤ (#‘𝑦) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦))))
70	67, 68, 69	syl2an 493	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (#‘𝑦) ↔ (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦))))
71	66, 70	mpbid 221	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)))
72	71	adantlr 747	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)))
73		breq2 4587	. . . . . . . . . . 11 ⊢ (((#‘𝐴) + (#‘𝑦)) = (#‘𝐵) → ((#‘𝐴) ≤ ((#‘𝐴) + (#‘𝑦)) ↔ (#‘𝐴) ≤ (#‘𝐵)))
74	72, 73	syl5ibcom 234	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((#‘𝐴) + (#‘𝑦)) = (#‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
75	63, 74	sylbid 229	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵)))
76	54, 75	sylan2 490	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +𝑜 (card‘𝑦))) = ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) → (#‘𝐴) ≤ (#‘𝐵)))
77	53, 76	syl5 33	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 (card‘𝑦)) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
78	52, 77	sylbird 249	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
79	78	rexlimdva 3013	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵) → (#‘𝐴) ≤ (#‘𝐵)))
80	48, 79	impbid 201	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
81		nnawordex 7604	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
82	26, 39, 81	syl2an 493	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +𝑜 𝑦) = (card‘𝐵)))
83		finnum 8657	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84		finnum 8657	. . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85		carddom2 8686	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
86	83, 84, 85	syl2an 493	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
87	80, 82, 86	3bitr2d 295	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵))
88	87	adantlr 747	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵))
89		hashxrcl 13010	. . . . . 6 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℝ*)
90	89	ad2antrr 758	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ∈ ℝ*)
91		pnfge 11840	. . . . 5 ⊢ ((#‘𝐴) ∈ ℝ* → (#‘𝐴) ≤ +∞)
92	90, 91	syl 17	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ +∞)
93		hashinf 12984	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
94	93	adantll 746	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐵) = +∞)
95	92, 94	breqtrrd 4611	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (#‘𝐴) ≤ (#‘𝐵))
96		isinffi 8701	. . . . . 6 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
97	96	ancoms 468	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
98	97	adantlr 747	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
99		brdomg 7851	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
100	99	ad2antlr 759	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
101	98, 100	mpbird 246	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴 ≼ 𝐵)
102	95, 101	2thd 254	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵))
103	88, 102	pm2.61dan 828	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((#‘𝐴) ≤ (#‘𝐵) ↔ 𝐴 ≼ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   ↾ cres 5040  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ωcom 6957  reccrdg 7392   +_𝑜 coa 7444   ≼ cdom 7839  Fincfn 7841  cardccrd 8644  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  +∞cpnf 9950  ℝ^*cxr 9952   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  ...cfz 12197  #chash 12979

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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  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-int 4411  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-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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980

This theorem is referenced by:  hashdomi  13030  hashsdom  13031  hashun2  13033  hashss  13058  hashsslei  13073  hashfun  13084  hashf1  13098  hashge3el3dif  13122  isercoll  14246  phicl2  15311  phibnd  15314  prmreclem2  15459  prmreclem3  15460  4sqlem11  15497  vdwlem11  15533  ramub2  15556  0ram  15562  ram0  15564  sylow1lem4  17839  pgpssslw  17852  fislw  17863  znfld  19728  znidomb  19729  fta1blem  23732  birthdaylem3  24480  basellem4  24610  ppiwordi  24688  musum  24717  ppiub  24729  chpub  24745  lgsqrlem4  24874  upgrex  25759  umgraex  25852  sizeusglecusg  26014  konigsberg  26514  derangenlem  30407  subfaclefac  30412  erdsze2lem1  30439  snmlff  30565  idomsubgmo  36795  sizusglecusg  40679  aacllem  42356