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Theorem omgadd 10911

Description: Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)

**Hypothesis**
Ref	Expression
omgadd.1	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)

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

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

Step	Hyp	Ref	Expression
1		oveq2 5933	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐴 +_o 𝑛) = (𝐴 +_o ∅))
2	1	fveq2d 5565	. . . . 5 ⊢ (𝑛 = ∅ → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o ∅)))
3		fveq2 5561	. . . . . 6 ⊢ (𝑛 = ∅ → (𝐺‘𝑛) = (𝐺‘∅))
4	3	oveq2d 5941	. . . . 5 ⊢ (𝑛 = ∅ → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘∅)))
5	2, 4	eqeq12d 2211	. . . 4 ⊢ (𝑛 = ∅ → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅))))
6	5	imbi2d 230	. . 3 ⊢ (𝑛 = ∅ → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))))
7		oveq2 5933	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝑧))
8	7	fveq2d 5565	. . . . 5 ⊢ (𝑛 = 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝑧)))
9		fveq2 5561	. . . . . 6 ⊢ (𝑛 = 𝑧 → (𝐺‘𝑛) = (𝐺‘𝑧))
10	9	oveq2d 5941	. . . . 5 ⊢ (𝑛 = 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑧)))
11	8, 10	eqeq12d 2211	. . . 4 ⊢ (𝑛 = 𝑧 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))))
12	11	imbi2d 230	. . 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 5933	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐴 +_o 𝑛) = (𝐴 +_o suc 𝑧))
14	13	fveq2d 5565	. . . . 5 ⊢ (𝑛 = suc 𝑧 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o suc 𝑧)))
15		fveq2 5561	. . . . . 6 ⊢ (𝑛 = suc 𝑧 → (𝐺‘𝑛) = (𝐺‘suc 𝑧))
16	15	oveq2d 5941	. . . . 5 ⊢ (𝑛 = suc 𝑧 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
17	14, 16	eqeq12d 2211	. . . 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 230	. . 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 5933	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐴 +_o 𝑛) = (𝐴 +_o 𝐵))
20	19	fveq2d 5565	. . . . 5 ⊢ (𝑛 = 𝐵 → (𝐺‘(𝐴 +_o 𝑛)) = (𝐺‘(𝐴 +_o 𝐵)))
21		fveq2 5561	. . . . . 6 ⊢ (𝑛 = 𝐵 → (𝐺‘𝑛) = (𝐺‘𝐵))
22	21	oveq2d 5941	. . . . 5 ⊢ (𝑛 = 𝐵 → ((𝐺‘𝐴) + (𝐺‘𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))
23	20, 22	eqeq12d 2211	. . . 4 ⊢ (𝑛 = 𝐵 → ((𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛)) ↔ (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
24	23	imbi2d 230	. . 3 ⊢ (𝑛 = 𝐵 → ((𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝑛)) = ((𝐺‘𝐴) + (𝐺‘𝑛))) ↔ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))))
25		omgadd.1	. . . . . . . . 9 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
26	25	frechashgf1o 10537	. . . . . . . 8 ⊢ 𝐺:ω–1-1-onto→ℕ₀
27		f1of 5507	. . . . . . . 8 ⊢ (𝐺:ω–1-1-onto→ℕ₀ → 𝐺:ω⟶ℕ₀)
28	26, 27	ax-mp 5	. . . . . . 7 ⊢ 𝐺:ω⟶ℕ₀
29	28	ffvelcdmi 5699	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℕ₀)
30	29	nn0cnd 9321	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐺‘𝐴) ∈ ℂ)
31	30	addridd 8192	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + 0) = (𝐺‘𝐴))
32		0zd 9355	. . . . . 6 ⊢ (𝐴 ∈ ω → 0 ∈ ℤ)
33	32, 25	frec2uz0d 10508	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐺‘∅) = 0)
34	33	oveq2d 5941	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐺‘𝐴) + (𝐺‘∅)) = ((𝐺‘𝐴) + 0))
35		nna0 6541	. . . . 5 ⊢ (𝐴 ∈ ω → (𝐴 +_o ∅) = 𝐴)
36	35	fveq2d 5565	. . . 4 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = (𝐺‘𝐴))
37	31, 34, 36	3eqtr4rd 2240	. . 3 ⊢ (𝐴 ∈ ω → (𝐺‘(𝐴 +_o ∅)) = ((𝐺‘𝐴) + (𝐺‘∅)))
38		nnasuc 6543	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o suc 𝑧) = suc (𝐴 +_o 𝑧))
39	38	fveq2d 5565	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = (𝐺‘suc (𝐴 +_o 𝑧)))
40		0zd 9355	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → 0 ∈ ℤ)
41		nnacl 6547	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐴 +_o 𝑧) ∈ ω)
42	40, 25, 41	frec2uzsucd 10510	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘suc (𝐴 +_o 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
43	39, 42	eqtrd 2229	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
44	43	3adant3 1019	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘(𝐴 +_o suc 𝑧)) = ((𝐺‘(𝐴 +_o 𝑧)) + 1))
45	30	3ad2ant1 1020	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘𝐴) ∈ ℂ)
46	28	ffvelcdmi 5699	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℕ₀)
47	46	nn0cnd 9321	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘𝑧) ∈ ℂ)
48	47	3ad2ant2 1021	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (𝐺‘𝑧) ∈ ℂ)
49		1cnd 8059	. . . . . . . . 9 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → 1 ∈ ℂ)
50	45, 48, 49	addassd 8066	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
51		oveq1 5932	. . . . . . . . 9 ⊢ ((𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧)) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
52	51	3ad2ant3 1022	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = (((𝐺‘𝐴) + (𝐺‘𝑧)) + 1))
53		0zd 9355	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → 0 ∈ ℤ)
54		id 19	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → 𝑧 ∈ ω)
55	53, 25, 54	frec2uzsucd 10510	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1))
56	55	oveq2d 5941	. . . . . . . . 9 ⊢ (𝑧 ∈ ω → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
57	56	3ad2ant2 1021	. . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘𝐴) + (𝐺‘suc 𝑧)) = ((𝐺‘𝐴) + ((𝐺‘𝑧) + 1)))
58	50, 52, 57	3eqtr4d 2239	. . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑧 ∈ ω ∧ (𝐺‘(𝐴 +_o 𝑧)) = ((𝐺‘𝐴) + (𝐺‘𝑧))) → ((𝐺‘(𝐴 +_o 𝑧)) + 1) = ((𝐺‘𝐴) + (𝐺‘suc 𝑧)))
59	44, 58	eqtrd 2229	. . . . . 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 1207	. . . . 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 116	. . . 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 26	. . 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, 37, 62	finds 4637	. 2 ⊢ (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵))))
64	63	impcom 125	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐺‘(𝐴 +_o 𝐵)) = ((𝐺‘𝐴) + (𝐺‘𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ∅c0 3451 ↦ cmpt 4095 suc csuc 4401 ωcom 4627 ⟶wf 5255 –1-1-onto→wf1o 5258 ‘cfv 5259 (class class class)co 5925 freccfrec 6457 +_o coa 6480 ℂcc 7894 0cc0 7896 1c1 7897 + caddc 7899 ℕ₀cn0 9266 ℤcz 9343

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-addcom 7996 ax-addass 7998 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-0id 8004 ax-rnegex 8005 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-ltadd 8012

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-oadd 6487 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-inn 9008 df-n0 9267 df-z 9344 df-uz 9619

This theorem is referenced by: hashun 10914

W3C validator