Theorem addsval2 27969

Description: The value of surreal addition with different choices for each bound variable. Definition from [Conway] p. 5. (Contributed by Scott Fenton, 21-Jan-2025.)

Assertion

Ref	Expression
addsval2	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)})))

Distinct variable groups:   𝐴,𝑙,𝑦   𝐴,𝑚,𝑧   𝐴,𝑟,𝑤   𝐴,𝑠,𝑡   𝐵,𝑙,𝑦   𝐵,𝑚,𝑧   𝐵,𝑟,𝑤   𝐵,𝑠,𝑡

Proof of Theorem addsval2

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addsval 27968	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏)}) |s ({𝑎 ∣ ∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑)})))
2		eqeq1 2741	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
3	2	rexbidv 3162	. . . . . 6 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ ( L ‘𝐴)𝑦 = (𝑏 +s 𝐵)))
4		oveq1 7367	. . . . . . . 8 ⊢ (𝑏 = 𝑙 → (𝑏 +s 𝐵) = (𝑙 +s 𝐵))
5	4	eqeq2d 2748	. . . . . . 7 ⊢ (𝑏 = 𝑙 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑙 +s 𝐵)))
6	5	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑏 ∈ ( L ‘𝐴)𝑦 = (𝑏 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵))
7	3, 6	bitrdi 287	. . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)))
8	7	cbvabv 2807	. . . 4 ⊢ {𝑎 ∣ ∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵)} = {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)}
9		eqeq1 2741	. . . . . . 7 ⊢ (𝑐 = 𝑧 → (𝑐 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑏)))
10	9	rexbidv 3162	. . . . . 6 ⊢ (𝑐 = 𝑧 → (∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑏 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑏)))
11		oveq2 7368	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝐴 +s 𝑏) = (𝐴 +s 𝑚))
12	11	eqeq2d 2748	. . . . . . 7 ⊢ (𝑏 = 𝑚 → (𝑧 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑚)))
13	12	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑏 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑏) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚))
14	10, 13	bitrdi 287	. . . . 5 ⊢ (𝑐 = 𝑧 → (∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)))
15	14	cbvabv 2807	. . . 4 ⊢ {𝑐 ∣ ∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏)} = {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}
16	8, 15	uneq12i 4107	. . 3 ⊢ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏)}) = ({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)})
17		eqeq1 2741	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑎 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑑 +s 𝐵)))
18	17	rexbidv 3162	. . . . . 6 ⊢ (𝑎 = 𝑤 → (∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑑 ∈ ( R ‘𝐴)𝑤 = (𝑑 +s 𝐵)))
19		oveq1 7367	. . . . . . . 8 ⊢ (𝑑 = 𝑟 → (𝑑 +s 𝐵) = (𝑟 +s 𝐵))
20	19	eqeq2d 2748	. . . . . . 7 ⊢ (𝑑 = 𝑟 → (𝑤 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑟 +s 𝐵)))
21	20	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑑 ∈ ( R ‘𝐴)𝑤 = (𝑑 +s 𝐵) ↔ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵))
22	18, 21	bitrdi 287	. . . . 5 ⊢ (𝑎 = 𝑤 → (∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)))
23	22	cbvabv 2807	. . . 4 ⊢ {𝑎 ∣ ∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵)} = {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)}
24		eqeq1 2741	. . . . . . 7 ⊢ (𝑐 = 𝑡 → (𝑐 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑑)))
25	24	rexbidv 3162	. . . . . 6 ⊢ (𝑐 = 𝑡 → (∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑑)))
26		oveq2 7368	. . . . . . . 8 ⊢ (𝑑 = 𝑠 → (𝐴 +s 𝑑) = (𝐴 +s 𝑠))
27	26	eqeq2d 2748	. . . . . . 7 ⊢ (𝑑 = 𝑠 → (𝑡 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑠)))
28	27	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑑 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑑) ↔ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠))
29	25, 28	bitrdi 287	. . . . 5 ⊢ (𝑐 = 𝑡 → (∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)))
30	29	cbvabv 2807	. . . 4 ⊢ {𝑐 ∣ ∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑)} = {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)}
31	23, 30	uneq12i 4107	. . 3 ⊢ ({𝑎 ∣ ∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑)}) = ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)})
32	16, 31	oveq12i 7372	. 2 ⊢ (({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏)}) |s ({𝑎 ∣ ∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑)})) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)}))
33	1, 32	eqtrdi 2788	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062   ∪ cun 3888  ‘cfv 6492  (class class class)co 7360   No csur 27617   |s ccuts 27765   L cleft 27831   R cright 27832   +_s cadds 27965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec2 27955  df-adds 27966

This theorem is referenced by:  addsproplem3  27977  leadds1  27995  addsuniflem  28007  addsasslem1  28009  addsasslem2  28010  addbday  28024  addsdilem1  28157