Theorem addsval2 27945

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 27944	. 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 3161	. . . . . 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 3216	. . . . . 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 3161	. . . . . 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 3216	. . . . . 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 4119	. . 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 3161	. . . . . 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 3216	. . . . . 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 3161	. . . . . 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 3216	. . . . . 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 4119	. . 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 3061   ∪ cun 3900  ‘cfv 6493  (class class class)co 7360   No csur 27611   |s cscut 27759   L cleft 27823   R cright 27824   +_s cadds 27941

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 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  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 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27614  df-slt 27615  df-bday 27616  df-sslt 27758  df-scut 27760  df-made 27825  df-old 27826  df-left 27828  df-right 27829  df-norec2 27931  df-adds 27942

This theorem is referenced by:  addsproplem3  27953  sleadd1  27971  addsuniflem  27983  addsasslem1  27985  addsasslem2  27986  addsbday  28000  addsdilem1  28133