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Theorem hashgadd 13027

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 6557	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐴 +_𝑜 𝑛) = (𝐴 +_𝑜 ∅))
2	1	fveq2d 6107	. . . . 5 ⊢ (𝑛 = ∅ → (𝐺‘(𝐴 +_𝑜 𝑛)) = (𝐺‘(𝐴 +_𝑜 ∅)))
3		fveq2 6103	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐺‘𝑛) = (𝐺‘∅))
4	3	oveq2d 6565	. . . . 5 ⊢ (𝑛 = ∅ → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘∅)))
5	2, 4	eqeq12d 2625	. . . 4 ⊢ (𝑛 = ∅ → ((𝐺‘(𝐴 +_𝑜 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_𝑜 ∅)) = ((𝐺‘𝐴) + (𝐺‘∅))))
6	5	imbi2d 329	. . 3 ⊢ (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_𝑜 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_𝑜 ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))))
7		oveq2 6557	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐴 +_𝑜 𝑛) = (𝐴 +_𝑜 𝑧))
8	7	fveq2d 6107	. . . . 5 ⊢ (𝑛 = 𝑧 → (𝐺‘(𝐴 +_𝑜 𝑛)) = (𝐺‘(𝐴 +_𝑜 𝑧)))
9		fveq2 6103	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
10	9	oveq2d 6565	. . . . 5 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))
11	8, 10	eqeq12d 2625	. . . 4 ⊢ (𝑛 = 𝑧 → ((𝐺‘(𝐴 +_𝑜 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_𝑜 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))))
12	11	imbi2d 329	. . 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 6557	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐴 +_𝑜 𝑛) = (𝐴 +_𝑜 suc 𝑧))
14	13	fveq2d 6107	. . . . 5 ⊢ (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +_𝑜 𝑛)) = (𝐺‘(𝐴 +_𝑜 suc 𝑧)))
15		fveq2 6103	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐺‘𝑛) = (𝐺‘suc 𝑧))
16	15	oveq2d 6565	. . . . 5 ⊢ (𝑛 = suc 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
17	14, 16	eqeq12d 2625	. . . 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 329	. . 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 6557	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐴 +_𝑜 𝑛) = (𝐴 +_𝑜 𝐵))
20	19	fveq2d 6107	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐺‘(𝐴 +_𝑜 𝑛)) = (𝐺‘(𝐴 +_𝑜 𝐵)))
21		fveq2 6103	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐺‘𝑛) = (𝐺‘𝐵))
22	21	oveq2d 6565	. . . . 5 ⊢ (𝑛 = 𝐵 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
23	20, 22	eqeq12d 2625	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐺‘(𝐴 +_𝑜 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_𝑜 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
24	23	imbi2d 329	. . 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 12632	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→ℕ₀
27		f1of 6050	. . . . . . . 8 ⊢ (𝐺:ω–1-1-onto→ℕ₀ → 𝐺:ω⟶ℕ₀)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ 𝐺:ω⟶ℕ₀
29	28	ffvelrni 6266	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℕ₀)
30	29	nn0cnd 11230	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℂ)
31	30	addid1d 10115	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + 0) = (𝐺‘𝐴))
32		0z 11265	. . . . . . 7 ⊢ 0 ∈ ℤ
33	32, 25	om2uz0i 12608	. . . . . 6 ⊢ (𝐺‘∅) = 0
34	33	oveq2i 6560	. . . . 5 ⊢ ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0)
35	34	a1i 11	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0))
36		nna0 7571	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 +_𝑜 ∅) = 𝐴)
37	36	fveq2d 6107	. . . 4 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_𝑜 ∅)) = (𝐺‘𝐴))
38	31, 35, 37	3eqtr4rd 2655	. . 3 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_𝑜 ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))
39		nnasuc 7573	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_𝑜 suc 𝑧) = suc (𝐴 +_𝑜 𝑧))
40	39	fveq2d 6107	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_𝑜 suc 𝑧)) = (𝐺‘suc (𝐴 +_𝑜 𝑧)))
41		nnacl 7578	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_𝑜 𝑧) ∈ ω)
42	32, 25	om2uzsuci 12609	. . . . . . . . . 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 2644	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_𝑜 suc 𝑧)) = ((𝐺‘(𝐴 +_𝑜 𝑧)) + 1))
45	44	3adant3 1074	. . . . . . 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	ffvelrni 6266	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℕ₀)
47	46	nn0cnd 11230	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℂ)
48		ax-1cn 9873	. . . . . . . . . . 11 ⊢ 1 ∈ ℂ
49		addass 9902	. . . . . . . . . . 11 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
50	48, 49	mp3an3 1405	. . . . . . . . . 10 ⊢ (((𝐺‘𝐴) ∈ ℂ ∧ (𝐺‘𝑧) ∈ ℂ) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
51	30, 47, 50	syl2an 493	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
52	51	3adant3 1074	. . . . . . . 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 6556	. . . . . . . . 9 ⊢ ((𝐺‘(𝐴 +_𝑜 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → ((𝐺‘(𝐴 +_𝑜 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
54	53	3ad2ant3 1077	. . . . . . . 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 12609	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1))
56	55	oveq2d 6565	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
57	56	3ad2ant2 1076	. . . . . . . 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 2654	. . . . . . 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 2644	. . . . . 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 1259	. . . . 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 450	. . . 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 6984	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +_𝑜 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
64	63	impcom 445	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +_𝑜 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ↦ cmpt 4643 ↾ cres 5040 suc csuc 5642 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ωcom 6957 reccrdg 7392 +_𝑜 coa 7444 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ℕ₀cn0 11169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564

This theorem is referenced by: hashdom 13029 hashun 13032

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