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Theorem hashgadd 14400

Description: 𝐺 maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

**Hypothesis**
Ref	Expression
hashgadd.1	⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)

**Assertion**
Ref	Expression
hashgadd	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))

Proof of Theorem hashgadd Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7418	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐴 +_o 𝑛) = (𝐴 +_o ∅))
2	1	fveq2d 6885	. . . . 5 ⊢ (𝑛 = ∅ → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o ∅)))
3		fveq2 6881	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐺‘𝑛) = (𝐺‘∅))
4	3	oveq2d 7426	. . . . 5 ⊢ (𝑛 = ∅ → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘∅)))
5	2, 4	eqeq12d 2752	. . . 4 ⊢ (𝑛 = ∅ → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅))))
6	5	imbi2d 340	. . 3 ⊢ (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))))
7		oveq2 7418	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝑧))
8	7	fveq2d 6885	. . . . 5 ⊢ (𝑛 = 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝑧)))
9		fveq2 6881	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
10	9	oveq2d 7426	. . . . 5 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))
11	8, 10	eqeq12d 2752	. . . 4 ⊢ (𝑛 = 𝑧 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))))
12	11	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))))
13		oveq2 7418	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o suc 𝑧))
14	13	fveq2d 6885	. . . . 5 ⊢ (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o suc 𝑧)))
15		fveq2 6881	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐺‘𝑛) = (𝐺‘suc 𝑧))
16	15	oveq2d 7426	. . . . 5 ⊢ (𝑛 = suc 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
17	14, 16	eqeq12d 2752	. . . 4 ⊢ (𝑛 = suc 𝑧 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))))
18	17	imbi2d 340	. . 3 ⊢ (𝑛 = suc 𝑧 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))))
19		oveq2 7418	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝐵))
20	19	fveq2d 6885	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝐵)))
21		fveq2 6881	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐺‘𝑛) = (𝐺‘𝐵))
22	21	oveq2d 7426	. . . . 5 ⊢ (𝑛 = 𝐵 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
23	20, 22	eqeq12d 2752	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
24	23	imbi2d 340	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))))
25		hashgadd.1	. . . . . . . . 9 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
26	25	hashgf1o 13994	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→ℕ₀
27		f1of 6823	. . . . . . . 8 ⊢ (𝐺:ω–1-1-onto→ℕ₀ → 𝐺:ω⟶ℕ₀)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ 𝐺:ω⟶ℕ₀
29	28	ffvelcdmi 7078	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℕ₀)
30	29	nn0cnd 12569	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℂ)
31	30	addridd 11440	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + 0) = (𝐺‘𝐴))
32		0z 12604	. . . . . . 7 ⊢ 0 ∈ ℤ
33	32, 25	om2uz0i 13970	. . . . . 6 ⊢ (𝐺‘∅) = 0
34	33	oveq2i 7421	. . . . 5 ⊢ ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0)
35	34	a1i 11	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0))
36		nna0 8621	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 +_o ∅) = 𝐴)
37	36	fveq2d 6885	. . . 4 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = (𝐺‘𝐴))
38	31, 35, 37	3eqtr4rd 2782	. . 3 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))
39		nnasuc 8623	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o suc 𝑧) = suc (𝐴 +_o 𝑧))
40	39	fveq2d 6885	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = (𝐺‘suc (𝐴 +_o 𝑧)))
41		nnacl 8628	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o 𝑧) ∈ ω)
42	32, 25	om2uzsuci 13971	. . . . . . . . . 10 ⊢ ((𝐴 +_o 𝑧) ∈ ω → (𝐺‘suc (𝐴 +_o 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
43	41, 42	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘suc (𝐴 +_o 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
44	40, 43	eqtrd 2771	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
45	44	3adant3 1132	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
46	28	ffvelcdmi 7078	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℕ₀)
47	46	nn0cnd 12569	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℂ)
48		ax-1cn 11192	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
49		addass 11221	. . . . . . . . . . 11 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
50	48, 49	mp3an3 1452	. . . . . . . . . 10 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
51	30, 47, 50	syl2an 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
52	51	3adant3 1132	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
53		oveq1 7417	. . . . . . . . 9 ⊢ ((𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
54	53	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
55	32, 25	om2uzsuci 13971	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1))
56	55	oveq2d 7426	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
57	56	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
58	52, 54, 57	3eqtr4d 2781	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
59	45, 58	eqtrd 2771	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
60	59	3expia 1121	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → ((𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧))))
61	60	expcom 413	. . . 4 ⊢ (𝑧 ∈ ω → (𝐴 ∈ ω → ((𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))))
62	61	a2d 29	. . 3 ⊢ (𝑧 ∈ ω → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐴 ∈ ω → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))))
63	6, 12, 18, 24, 38, 62	finds 7897	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
64	63	impcom 407	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 Vcvv 3464 ∅c0 4313 ↦ cmpt 5206 ↾ cres 5661 suc csuc 6359 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7410 ωcom 7866 reccrdg 8428 +_o coa 8482 ℂcc 11132 0cc0 11134 1c1 11135 + caddc 11137 ℕ₀cn0 12506

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-oadd 8489 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-n0 12507 df-z 12594 df-uz 12858

This theorem is referenced by: hashdom 14402 hashun 14405

W3C validator