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Theorem hashgadd 14415

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 7420	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐴 +_o 𝑛) = (𝐴 +_o ∅))
2	1	fveq2d 6887	. . . . 5 ⊢ (𝑛 = ∅ → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o ∅)))
3		fveq2 6883	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐺‘𝑛) = (𝐺‘∅))
4	3	oveq2d 7428	. . . . 5 ⊢ (𝑛 = ∅ → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘∅)))
5	2, 4	eqeq12d 2779	. . . 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 7420	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝑧))
8	7	fveq2d 6887	. . . . 5 ⊢ (𝑛 = 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝑧)))
9		fveq2 6883	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
10	9	oveq2d 7428	. . . . 5 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))
11	8, 10	eqeq12d 2779	. . . 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 7420	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o suc 𝑧))
14	13	fveq2d 6887	. . . . 5 ⊢ (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o suc 𝑧)))
15		fveq2 6883	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐺‘𝑛) = (𝐺‘suc 𝑧))
16	15	oveq2d 7428	. . . . 5 ⊢ (𝑛 = suc 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
17	14, 16	eqeq12d 2779	. . . 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 7420	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝐵))
20	19	fveq2d 6887	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝐵)))
21		fveq2 6883	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐺‘𝑛) = (𝐺‘𝐵))
22	21	oveq2d 7428	. . . . 5 ⊢ (𝑛 = 𝐵 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
23	20, 22	eqeq12d 2779	. . . 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 14009	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→ℕ₀
27		f1of 6822	. . . . . . . 8 ⊢ (𝐺:ω–1-1-onto→ℕ₀ → 𝐺:ω⟶ℕ₀)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ 𝐺:ω⟶ℕ₀
29	28	ffvelcdmi 7080	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℕ₀)
30	29	nn0cnd 12568	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℂ)
31	30	addridd 11411	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + 0) = (𝐺‘𝐴))
32		0z 12603	. . . . . . 7 ⊢ 0 ∈ ℤ
33	32, 25	om2uz0i 13985	. . . . . 6 ⊢ (𝐺‘∅) = 0
34	33	oveq2i 7423	. . . . 5 ⊢ ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0)
35	34	a1i 11	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0))
36		nna0 8591	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 +_o ∅) = 𝐴)
37	36	fveq2d 6887	. . . 4 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = (𝐺‘𝐴))
38	31, 35, 37	3eqtr4rd 2809	. . 3 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))
39		nnasuc 8593	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o suc 𝑧) = suc (𝐴 +_o 𝑧))
40	39	fveq2d 6887	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = (𝐺‘suc (𝐴 +_o 𝑧)))
41		nnacl 8598	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o 𝑧) ∈ ω)
42	32, 25	om2uzsuci 13986	. . . . . . . . . 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 2798	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
45	44	3adant3 1150	. . . . . . 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 7080	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℕ₀)
47	46	nn0cnd 12568	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℂ)
48		ax-1cn 11159	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
49		addass 11188	. . . . . . . . . . 11 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
50	48, 49	mp3an3 1479	. . . . . . . . . 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 1150	. . . . . . . 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 7419	. . . . . . . . 9 ⊢ ((𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
54	53	3ad2ant3 1153	. . . . . . . 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 13986	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1))
56	55	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
57	56	3ad2ant2 1152	. . . . . . . 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 2808	. . . . . . 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 2798	. . . . . 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 1139	. . . . 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 7894	. 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 1103 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ∅c0 4287 ↦ cmpt 5193 ↾ cres 5665 suc csuc 6364 ⟶wf 6534 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 ωcom 7863 reccrdg 8397 +_o coa 8451 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 ℕ₀cn0 12505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-oadd 8458 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-nn 12235 df-n0 12506 df-z 12593 df-uz 12864

This theorem is referenced by: hashdom 14417 hashun 14420 hashnna 45711

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