hashdom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hashdom		Structured version Visualization version GIF version

Theorem hashdom 14403

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 13996	. . . . . . . 8 ⊢ (1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin
2		ficardom 9935	. . . . . . . 8 ⊢ ((1...((♯‘𝐵) − (♯‘𝐴))) ∈ Fin → (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω)
3	1, 2	ax-mp 5	. . . . . . 7 ⊢ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω
4		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω) = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
5	4	hashgval 14357	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
6	5	ad2antrr 738	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
7	4	hashgval 14357	. . . . . . . . . . . . . 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 14380	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
10	9	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ∈ ℕ₀)
11		hashcl 14380	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ₀)
12	11	ad2antlr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐵) ∈ ℕ₀)
13		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘𝐴) ≤ (♯‘𝐵))
14		nn0sub2 12645	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
15	10, 12, 13, 14	syl3anc 1394	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀)
16		hashfz1 14370	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐵) − (♯‘𝐴)) ∈ ℕ₀ → (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))) = ((♯‘𝐵) − (♯‘𝐴)))
17	15, 16	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (♯‘(1...((♯‘𝐵) − (♯‘𝐴)))) = ((♯‘𝐵) − (♯‘𝐴)))
18	8, 17	eqtrid 2812	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = ((♯‘𝐵) − (♯‘𝐴)))
19	6, 18	oveq12d 7418	. . . . . . . . . . 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 12555	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
21	11	nn0cnd 12555	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℂ)
22		pncan3 11453	. . . . . . . . . . . . 13 ⊢ (((♯‘𝐴) ∈ ℂ ∧ (♯‘𝐵) ∈ ℂ) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
23	20, 21, 22	syl2an 607	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
24	23	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((♯‘𝐴) + ((♯‘𝐵) − (♯‘𝐴))) = (♯‘𝐵))
25	19, 24	eqtrd 2800	. . . . . . . . . 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 9935	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
27	26	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐴) ∈ ω)
28	4	hashgadd 14401	. . . . . . . . . . 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 597	. . . . . . . . . 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 14357	. . . . . . . . . . 11 ⊢ (𝐵 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
31	30	ad2antlr 739	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
32	25, 29, 31	3eqtr4d 2810	. . . . . . . . 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 6875	. . . . . . . 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 13995	. . . . . . . . 9 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω):ω–1-1-onto→ℕ₀
35		nnacl 8585	. . . . . . . . . 10 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) ∈ ω) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
36	27, 3, 35	sylancl 597	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) ∈ ω)
37		f1ocnvfv1 7264	. . . . . . . . 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 598	. . . . . . . 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 9935	. . . . . . . . . 10 ⊢ (𝐵 ∈ Fin → (card‘𝐵) ∈ ω)
40	39	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → (card‘𝐵) ∈ ω)
41		f1ocnvfv1 7264	. . . . . . . . 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 598	. . . . . . . 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 2808	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵))
44		oveq2 7408	. . . . . . . . 9 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → ((card‘𝐴) +_o 𝑦) = ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))))
45	44	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑦 = (card‘(1...((♯‘𝐵) − (♯‘𝐴)))) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) ↔ ((card‘𝐴) +_o (card‘(1...((♯‘𝐵) − (♯‘𝐴))))) = (card‘𝐵)))
46	45	rspcev 3584	. . . . . . 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 598	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ (♯‘𝐴) ≤ (♯‘𝐵)) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵))
48	47	ex 417	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) → ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
49		cardnn 9937	. . . . . . . . . 10 ⊢ (𝑦 ∈ ω → (card‘𝑦) = 𝑦)
50	49	adantl 486	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (card‘𝑦) = 𝑦)
51	50	oveq2d 7416	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → ((card‘𝐴) +_o (card‘𝑦)) = ((card‘𝐴) +_o 𝑦))
52	51	eqeq1d 2767	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) ↔ ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
53		fveq2 6871	. . . . . . . 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 9140	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → 𝑦 ∈ Fin)
55		ficardom 9935	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (card‘𝑦) ∈ ω)
56	4	hashgadd 14401	. . . . . . . . . . . . . 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 607	. . . . . . . . . . . . 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 14357	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝑦)) = (♯‘𝑦))
59	5, 58	oveqan12d 7419	. . . . . . . . . . . . 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 2800	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
61	60	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘((card‘𝐴) +_o (card‘𝑦))) = ((♯‘𝐴) + (♯‘𝑦)))
62	30	ad2antlr 739	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → ((rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)‘(card‘𝐵)) = (♯‘𝐵))
63	61, 62	eqeq12d 2781	. . . . . . . . . 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 14380	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
65	64	nn0ge0d 12556	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → 0 ≤ (♯‘𝑦))
66	65	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → 0 ≤ (♯‘𝑦))
67	9	nn0red 12554	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ)
68	64	nn0red 12554	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℝ)
69		addge01 11712	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝑦) ∈ ℝ) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
70	67, 68, 69	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (0 ≤ (♯‘𝑦) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦))))
71	66, 70	mpbid 235	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
72	71	adantlr 727	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)))
73		breq2 5108	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → ((♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝑦)) ↔ (♯‘𝐴) ≤ (♯‘𝐵)))
74	72, 73	syl5ibcom 248	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ Fin) → (((♯‘𝐴) + (♯‘𝑦)) = (♯‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
75	63, 74	sylbid 243	. . . . . . . . 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 604	. . . . . . . 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 35	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o (card‘𝑦)) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
78	52, 77	sylbird 263	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) ∧ 𝑦 ∈ ω) → (((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
79	78	rexlimdva 3166	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵) → (♯‘𝐴) ≤ (♯‘𝐵)))
80	48, 79	impbid 215	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
81		nnawordex 8611	. . . . 5 ⊢ (((card‘𝐴) ∈ ω ∧ (card‘𝐵) ∈ ω) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
82	26, 39, 81	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ∃𝑦 ∈ ω ((card‘𝐴) +_o 𝑦) = (card‘𝐵)))
83		finnum 9922	. . . . 5 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ dom card)
84		finnum 9922	. . . . 5 ⊢ (𝐵 ∈ Fin → 𝐵 ∈ dom card)
85		carddom2 9951	. . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
86	83, 84, 85	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵))
87	80, 82, 86	3bitr2d 310	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
88	87	adantlr 727	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
89		hashxrcl 14381	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℝ^*)
90	89	ad2antrr 738	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ∈ ℝ^*)
91		pnfge 13143	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℝ^* → (♯‘𝐴) ≤ +∞)
92	90, 91	syl 18	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ +∞)
93		hashinf 14359	. . . . 5 ⊢ ((𝐵 ∈ 𝑉 ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
94	93	adantll 726	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐵) = +∞)
95	92, 94	breqtrrd 5132	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (♯‘𝐴) ≤ (♯‘𝐵))
96		isinffi 9966	. . . . . 6 ⊢ ((¬ 𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
97	96	ancoms 463	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
98	97	adantlr 727	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ∃𝑓 𝑓:𝐴–1-1→𝐵)
99		brdomg 8943	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
100	99	ad2antlr 739	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵))
101	98, 100	mpbird 260	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → 𝐴 ≼ 𝐵)
102	95, 101	2thd 268	. 2 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) ∧ ¬ 𝐵 ∈ Fin) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))
103	88, 102	pm2.61dan 824	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ 𝐴 ≼ 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1563 ∃wex 1802 ∈ wcel 2145 ∃wrex 3089 Vcvv 3457 ⊆ wss 3907 class class class wbr 5104 ↦ cmpt 5185 ^◡ccnv 5650 dom cdm 5651 ↾ cres 5653 –1-1→wf1 6522 –1-1-onto→wf1o 6524 ‘cfv 6525 (class class class)co 7400 ωcom 7850 reccrdg 8384 +_o coa 8438 ≼ cdom 8929 Fincfn 8931 cardccrd 9909 ℂcc 11086 ℝcr 11087 0cc0 11088 1c1 11089 + caddc 11091 +∞cpnf 11228 ℝ^*cxr 11230 ≤ cle 11232 − cmin 11429 ℕ₀cn0 12492 ...cfz 13523 ♯chash 14354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-oadd 8445 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12222 df-n0 12493 df-xnn0 12566 df-z 12580 df-uz 12851 df-fz 13524 df-hash 14355

This theorem is referenced by: hashdomi 14404 hashsdom 14405 hashun2 14407 hashss 14433 hashsslei 14451 hashfun 14462 hashf1 14482 hashge3el3dif 14512 isercoll 15707 phicl2 16815 phibnd 16818 prmreclem2 16965 prmreclem3 16966 4sqlem11 17003 vdwlem11 17039 ramub2 17062 0ram 17068 ram0 17070 sylow1lem4 19659 pgpssslw 19672 fislw 19683 znfld 21667 znidomb 21668 fta1blem 26285 birthdaylem3 27072 basellem4 27202 ppiwordi 27280 musum 27309 ppiub 27322 chpub 27338 lgsqrlem4 27467 upgrex 29347 sizusglecusg 29718 derangenlem 35529 subfaclefac 35534 erdsze2lem1 35561 snmlff 35687 hashnexinj 42752 idomsubgmo 43777 aacllem 50431

W3C validator