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Theorem hashgadd 14409

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 7416	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐴 +_o 𝑛) = (𝐴 +_o ∅))
2	1	fveq2d 6883	. . . . 5 ⊢ (𝑛 = ∅ → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o ∅)))
3		fveq2 6879	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐺‘𝑛) = (𝐺‘∅))
4	3	oveq2d 7424	. . . . 5 ⊢ (𝑛 = ∅ → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘∅)))
5	2, 4	eqeq12d 2785	. . . 4 ⊢ (𝑛 = ∅ → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅))))
6	5	imbi2d 343	. . 3 ⊢ (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))))
7		oveq2 7416	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝑧))
8	7	fveq2d 6883	. . . . 5 ⊢ (𝑛 = 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝑧)))
9		fveq2 6879	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
10	9	oveq2d 7424	. . . . 5 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))
11	8, 10	eqeq12d 2785	. . . 4 ⊢ (𝑛 = 𝑧 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))))
12	11	imbi2d 343	. . 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 7416	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o suc 𝑧))
14	13	fveq2d 6883	. . . . 5 ⊢ (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o suc 𝑧)))
15		fveq2 6879	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐺‘𝑛) = (𝐺‘suc 𝑧))
16	15	oveq2d 7424	. . . . 5 ⊢ (𝑛 = suc 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
17	14, 16	eqeq12d 2785	. . . 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 343	. . 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 7416	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝐵))
20	19	fveq2d 6883	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝐵)))
21		fveq2 6879	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐺‘𝑛) = (𝐺‘𝐵))
22	21	oveq2d 7424	. . . . 5 ⊢ (𝑛 = 𝐵 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
23	20, 22	eqeq12d 2785	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
24	23	imbi2d 343	. . 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 14003	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→ℕ₀
27		f1of 6818	. . . . . . . 8 ⊢ (𝐺:ω–1-1-onto→ℕ₀ → 𝐺:ω⟶ℕ₀)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ 𝐺:ω⟶ℕ₀
29	28	ffvelcdmi 7076	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℕ₀)
30	29	nn0cnd 12563	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℂ)
31	30	addridd 11406	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + 0) = (𝐺‘𝐴))
32		0z 12598	. . . . . . 7 ⊢ 0 ∈ ℤ
33	32, 25	om2uz0i 13979	. . . . . 6 ⊢ (𝐺‘∅) = 0
34	33	oveq2i 7419	. . . . 5 ⊢ ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0)
35	34	a1i 11	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0))
36		nna0 8586	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 +_o ∅) = 𝐴)
37	36	fveq2d 6883	. . . 4 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = (𝐺‘𝐴))
38	31, 35, 37	3eqtr4rd 2815	. . 3 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))
39		nnasuc 8588	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o suc 𝑧) = suc (𝐴 +_o 𝑧))
40	39	fveq2d 6883	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = (𝐺‘suc (𝐴 +_o 𝑧)))
41		nnacl 8593	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o 𝑧) ∈ ω)
42	32, 25	om2uzsuci 13980	. . . . . . . . . 10 ⊢ ((𝐴 +_o 𝑧) ∈ ω → (𝐺‘suc (𝐴 +_o 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
43	41, 42	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘suc (𝐴 +_o 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
44	40, 43	eqtrd 2804	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
45	44	3adant3 1148	. . . . . . 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 7076	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℕ₀)
47	46	nn0cnd 12563	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℂ)
48		ax-1cn 11154	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
49		addass 11183	. . . . . . . . . . 11 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
50	48, 49	mp3an3 1476	. . . . . . . . . 10 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
51	30, 47, 50	syl2an 607	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
52	51	3adant3 1148	. . . . . . . 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 7415	. . . . . . . . 9 ⊢ ((𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
54	53	3ad2ant3 1151	. . . . . . . 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 13980	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1))
56	55	oveq2d 7424	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
57	56	3ad2ant2 1150	. . . . . . . 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 2814	. . . . . . 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 2804	. . . . . 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 1137	. . . . 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 418	. . . 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 30	. . 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 7889	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
64	63	impcom 412	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ∅c0 4294 ↦ cmpt 5193 ↾ cres 5661 suc csuc 6359 ⟶wf 6529 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7408 ωcom 7858 reccrdg 8392 +_o coa 8446 ℂcc 11094 0cc0 11096 1c1 11097 + caddc 11099 ℕ₀cn0 12500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 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 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-n0 12501 df-z 12588 df-uz 12859

This theorem is referenced by: hashdom 14411 hashun 14414 hashnna 45613

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