Theorem addsval2 28350

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 28349	. 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 2764	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
3	2	rexbidv 3186	. . . . . 6 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ (L‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ (L‘𝐴)𝑦 = (𝑏 +s 𝐵)))
4		oveq1 7427	. . . . . . . 8 ⊢ (𝑏 = 𝑙 → (𝑏 +s 𝐵) = (𝑙 +s 𝐵))
5	4	eqeq2d 2771	. . . . . . 7 ⊢ (𝑏 = 𝑙 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑙 +s 𝐵)))
6	5	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑏 ∈ (L‘𝐴)𝑦 = (𝑏 +s 𝐵) ↔ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵))
7	3, 6	bitrdi 290	. . . . 5 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ (L‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)))
8	7	cbvabv 2830	. . . 4 ⊢ {𝑎 ∣ ∃𝑏 ∈ (L‘𝐴)𝑎 = (𝑏 +s 𝐵)} = {𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)}
9		eqeq1 2764	. . . . . . 7 ⊢ (𝑐 = 𝑧 → (𝑐 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑏)))
10	9	rexbidv 3186	. . . . . 6 ⊢ (𝑐 = 𝑧 → (∃𝑏 ∈ (L‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑏 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑏)))
11		oveq2 7428	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝐴 +s 𝑏) = (𝐴 +s 𝑚))
12	11	eqeq2d 2771	. . . . . . 7 ⊢ (𝑏 = 𝑚 → (𝑧 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑚)))
13	12	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑏 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑏) ↔ ∃𝑚 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑚))
14	10, 13	bitrdi 290	. . . . 5 ⊢ (𝑐 = 𝑧 → (∃𝑏 ∈ (L‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑚 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑚)))
15	14	cbvabv 2830	. . . 4 ⊢ {𝑐 ∣ ∃𝑏 ∈ (L‘𝐵)𝑐 = (𝐴 +s 𝑏)} = {𝑧 ∣ ∃𝑚 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑚)}
16	8, 15	uneq12i 4165	. . 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 2764	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑎 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑑 +s 𝐵)))
18	17	rexbidv 3186	. . . . . 6 ⊢ (𝑎 = 𝑤 → (∃𝑑 ∈ (R‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑑 ∈ (R‘𝐴)𝑤 = (𝑑 +s 𝐵)))
19		oveq1 7427	. . . . . . . 8 ⊢ (𝑑 = 𝑟 → (𝑑 +s 𝐵) = (𝑟 +s 𝐵))
20	19	eqeq2d 2771	. . . . . . 7 ⊢ (𝑑 = 𝑟 → (𝑤 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑟 +s 𝐵)))
21	20	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑑 ∈ (R‘𝐴)𝑤 = (𝑑 +s 𝐵) ↔ ∃𝑟 ∈ (R‘𝐴)𝑤 = (𝑟 +s 𝐵))
22	18, 21	bitrdi 290	. . . . 5 ⊢ (𝑎 = 𝑤 → (∃𝑑 ∈ (R‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑟 ∈ (R‘𝐴)𝑤 = (𝑟 +s 𝐵)))
23	22	cbvabv 2830	. . . 4 ⊢ {𝑎 ∣ ∃𝑑 ∈ (R‘𝐴)𝑎 = (𝑑 +s 𝐵)} = {𝑤 ∣ ∃𝑟 ∈ (R‘𝐴)𝑤 = (𝑟 +s 𝐵)}
24		eqeq1 2764	. . . . . . 7 ⊢ (𝑐 = 𝑡 → (𝑐 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑑)))
25	24	rexbidv 3186	. . . . . 6 ⊢ (𝑐 = 𝑡 → (∃𝑑 ∈ (R‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑑)))
26		oveq2 7428	. . . . . . . 8 ⊢ (𝑑 = 𝑠 → (𝐴 +s 𝑑) = (𝐴 +s 𝑠))
27	26	eqeq2d 2771	. . . . . . 7 ⊢ (𝑑 = 𝑠 → (𝑡 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑠)))
28	27	cbvrexvw 3241	. . . . . 6 ⊢ (∃𝑑 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑑) ↔ ∃𝑠 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑠))
29	25, 28	bitrdi 290	. . . . 5 ⊢ (𝑐 = 𝑡 → (∃𝑑 ∈ (R‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑠 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑠)))
30	29	cbvabv 2830	. . . 4 ⊢ {𝑐 ∣ ∃𝑑 ∈ (R‘𝐵)𝑐 = (𝐴 +s 𝑑)} = {𝑡 ∣ ∃𝑠 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑠)}
31	23, 30	uneq12i 4165	. . 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 7432	. 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 2811	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)})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ∪ cun 4149  ‘cfv 6538  (class class class)co 7420  Nocsur 27998   |s ccuts 28146  Lcleft 28212  Rcright 28213   +_s cadds 28346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-1o 8477  df-2o 8478  df-no 28001  df-lts 28002  df-bday 28003  df-slts 28145  df-cuts 28147  df-made 28214  df-old 28215  df-left 28217  df-right 28218  df-norec2 28336  df-adds 28347

This theorem is used by:  addsproplem3  28358  leadds1  28376  addsuniflem  28388  addsasslem1  28390  addsasslem2  28391  addbday  28405  addsdilem1  28538