Theorem addsval2 28379

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 28378	. 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 2767	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
3	2	rexbidv 3190	. . . . . 6 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ (L‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ (L‘𝐴)𝑦 = (𝑏 +s 𝐵)))
4		oveq1 7450	. . . . . . . 8 ⊢ (𝑏 = 𝑙 → (𝑏 +s 𝐵) = (𝑙 +s 𝐵))
5	4	eqeq2d 2774	. . . . . . 7 ⊢ (𝑏 = 𝑙 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑙 +s 𝐵)))
6	5	cbvrexvw 3245	. . . . . 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 2834	. . . 4 ⊢ {𝑎 ∣ ∃𝑏 ∈ (L‘𝐴)𝑎 = (𝑏 +s 𝐵)} = {𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)}
9		eqeq1 2767	. . . . . . 7 ⊢ (𝑐 = 𝑧 → (𝑐 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑏)))
10	9	rexbidv 3190	. . . . . 6 ⊢ (𝑐 = 𝑧 → (∃𝑏 ∈ (L‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑏 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑏)))
11		oveq2 7451	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝐴 +s 𝑏) = (𝐴 +s 𝑚))
12	11	eqeq2d 2774	. . . . . . 7 ⊢ (𝑏 = 𝑚 → (𝑧 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑚)))
13	12	cbvrexvw 3245	. . . . . 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 2834	. . . 4 ⊢ {𝑐 ∣ ∃𝑏 ∈ (L‘𝐵)𝑐 = (𝐴 +s 𝑏)} = {𝑧 ∣ ∃𝑚 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑚)}
16	8, 15	uneq12i 4168	. . 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 2767	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑎 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑑 +s 𝐵)))
18	17	rexbidv 3190	. . . . . 6 ⊢ (𝑎 = 𝑤 → (∃𝑑 ∈ (R‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑑 ∈ (R‘𝐴)𝑤 = (𝑑 +s 𝐵)))
19		oveq1 7450	. . . . . . . 8 ⊢ (𝑑 = 𝑟 → (𝑑 +s 𝐵) = (𝑟 +s 𝐵))
20	19	eqeq2d 2774	. . . . . . 7 ⊢ (𝑑 = 𝑟 → (𝑤 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑟 +s 𝐵)))
21	20	cbvrexvw 3245	. . . . . 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 2834	. . . 4 ⊢ {𝑎 ∣ ∃𝑑 ∈ (R‘𝐴)𝑎 = (𝑑 +s 𝐵)} = {𝑤 ∣ ∃𝑟 ∈ (R‘𝐴)𝑤 = (𝑟 +s 𝐵)}
24		eqeq1 2767	. . . . . . 7 ⊢ (𝑐 = 𝑡 → (𝑐 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑑)))
25	24	rexbidv 3190	. . . . . 6 ⊢ (𝑐 = 𝑡 → (∃𝑑 ∈ (R‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑑)))
26		oveq2 7451	. . . . . . . 8 ⊢ (𝑑 = 𝑠 → (𝐴 +s 𝑑) = (𝐴 +s 𝑠))
27	26	eqeq2d 2774	. . . . . . 7 ⊢ (𝑑 = 𝑠 → (𝑡 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑠)))
28	27	cbvrexvw 3245	. . . . . 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 2834	. . . 4 ⊢ {𝑐 ∣ ∃𝑑 ∈ (R‘𝐵)𝑐 = (𝐴 +s 𝑑)} = {𝑡 ∣ ∃𝑠 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑠)}
31	23, 30	uneq12i 4168	. . 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 7455	. 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 2814	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 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ∪ cun 4152  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   |s ccuts 28175  Lcleft 28241  Rcright 28242   +_s cadds 28375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec2 28365  df-adds 28376

This theorem is used by:  addsproplem3  28387  leadds1  28405  addsuniflem  28417  addsasslem1  28419  addsasslem2  28420  addbday  28434  addsdilem1  28567