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Theorem hashdom 14302

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 13895	. . . . . . . 8 ⊢ (1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin
2		ficardom 9873	. . . . . . . 8 ⊢ ((1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin → (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω)
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω
4		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
5	4	hashgval 14256	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
6	5	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
7	4	hashgval 14256	. . . . . . . . . . . . . 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 14279	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
10	9	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ∈ ℕ₀)
11		hashcl 14279	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
12	11	ad2antlr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐵) ∈ ℕ₀)
13		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵))
14		nn0sub2 12553	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
15	10, 12, 13, 14	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
16		hashfz1 14269	. . . . . . . . . . . . . 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	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = ((♯‘𝐵) − (♯‘𝐴)))
19	6, 18	oveq12d 7376	. . . . . . . . . . 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 12464	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
21	11	nn0cnd 12464	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
22		pncan3 11388	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
23	20, 21, 22	syl2an 596	. . . . . . . . . . . 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 2771	. . . . . . . . . 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 9873	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
27	26	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐴) ∈ ω)
28	4	hashgadd 14300	. . . . . . . . . . 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 586	. . . . . . . . . 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 14256	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
31	30	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
32	25, 29, 31	3eqtr4d 2781	. . . . . . . . 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 6838	. . . . . . . 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 13894	. . . . . . . . 9 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀
35		nnacl 8539	. . . . . . . . . 10 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
36	27, 3, 35	sylancl 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
37		f1ocnvfv1 7222	. . . . . . . . 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 587	. . . . . . . 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 9873	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
40	39	ad2antlr 727	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐵) ∈ ω)
41		f1ocnvfv1 7222	. . . . . . . . 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 587	. . . . . . . 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 2779	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵))
44		oveq2 7366	. . . . . . . . 9 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → ((card‘𝐴) +_o 𝑦) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
45	44	eqeq1d 2738	. . . . . . . 8 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵)))
46	45	rspcev 3576	. . . . . . 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 587	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵))
48	47	ex 412	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
49		cardnn 9875	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
50	49	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
51	50	oveq2d 7374	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +_o (card‘𝑦)) = ((card‘𝐴) +_o 𝑦))
52	51	eqeq1d 2738	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
53		fveq2 6834	. . . . . . . 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 9092	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ Fin)
55		ficardom 9873	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
56	4	hashgadd 14300	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 14256	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (♯‘𝑦))
59	5, 58	oveqan12d 7377	. . . . . . . . . . . . 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 2771	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
61	60	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
62	30	ad2antlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
63	61, 62	eqeq12d 2752	. . . . . . . . . 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 14279	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
65	64	nn0ge0d 12465	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → 0 ≤ (♯‘𝑦))
66	65	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (♯‘𝑦))
67	9	nn0red 12463	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
68	64	nn0red 12463	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℝ)
69		addge01 11647	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝑦) ∈ ℝ) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
70	67, 68, 69	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
71	66, 70	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
72	71	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
73		breq2 5102	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → ((♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)) ↔ (♯‘𝐴) ≤ (♯‘𝐵)))
74	72, 73	syl5ibcom 245	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
75	63, 74	sylbid 240	. . . . . . . . 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 593	. . . . . . . 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 34	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
78	52, 77	sylbird 260	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
79	78	rexlimdva 3137	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
80	48, 79	impbid 212	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
81		nnawordex 8565	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
82	26, 39, 81	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
83		finnum 9860	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84		finnum 9860	. . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85		carddom2 9889	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
86	83, 84, 85	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
87	80, 82, 86	3bitr2d 307	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
88	87	adantlr 715	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
89		hashxrcl 14280	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ^*)
90	89	ad2antrr 726	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ∈ ℝ^*)
91		pnfge 13044	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℝ^* → (♯‘𝐴) ≤ +∞)
92	90, 91	syl 17	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ +∞)
93		hashinf 14258	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
94	93	adantll 714	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
95	92, 94	breqtrrd 5126	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵))
96		isinffi 9904	. . . . . 6 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
97	96	ancoms 458	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
98	97	adantlr 715	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
99		brdomg 8895	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
100	99	ad2antlr 727	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
101	98, 100	mpbird 257	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴 ≼ 𝐵)
102	95, 101	2thd 265	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
103	88, 102	pm2.61dan 812	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∃wex 1780 ∈ wcel 2113 ∃wrex 3060 Vcvv 3440 ⊆ wss 3901 class class class wbr 5098 ↦ cmpt 5179 ^◡ccnv 5623 dom cdm 5624 ↾ cres 5626 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 ωcom 7808 reccrdg 8340 +_o coa 8394 ≼ cdom 8881 Fincfn 8883 cardccrd 9847 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 + caddc 11029 +∞cpnf 11163 ℝ^*cxr 11165 ≤ cle 11167 − cmin 11364 ℕ₀cn0 12401 ...cfz 13423 ♯chash 14253

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-oadd 8401 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-n0 12402 df-xnn0 12475 df-z 12489 df-uz 12752 df-fz 13424 df-hash 14254

This theorem is referenced by: hashdomi 14303 hashsdom 14304 hashun2 14306 hashss 14332 hashsslei 14349 hashfun 14360 hashf1 14380 hashge3el3dif 14410 isercoll 15591 phicl2 16695 phibnd 16698 prmreclem2 16845 prmreclem3 16846 4sqlem11 16883 vdwlem11 16919 ramub2 16942 0ram 16948 ram0 16950 sylow1lem4 19530 pgpssslw 19543 fislw 19554 znfld 21515 znidomb 21516 fta1blem 26132 birthdaylem3 26919 basellem4 27050 ppiwordi 27128 musum 27157 ppiub 27171 chpub 27187 lgsqrlem4 27316 upgrex 29165 sizusglecusg 29537 derangenlem 35365 subfaclefac 35370 erdsze2lem1 35397 snmlff 35523 hashnexinj 42382 idomsubgmo 43435 aacllem 50046

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