Theorem addsval2 28349

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 28348	. 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 2765	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
3	2	rexbidv 3187	. . . . . 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 2772	. . . . . . 7 ⊢ (𝑏 = 𝑙 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑙 +s 𝐵)))
6	5	cbvrexvw 3242	. . . . . 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 2831	. . . 4 ⊢ {𝑎 ∣ ∃𝑏 ∈ (L‘𝐴)𝑎 = (𝑏 +s 𝐵)} = {𝑦 ∣ ∃𝑙 ∈ (L‘𝐴)𝑦 = (𝑙 +s 𝐵)}
9		eqeq1 2765	. . . . . . 7 ⊢ (𝑐 = 𝑧 → (𝑐 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑏)))
10	9	rexbidv 3187	. . . . . 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 2772	. . . . . . 7 ⊢ (𝑏 = 𝑚 → (𝑧 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑚)))
13	12	cbvrexvw 3242	. . . . . 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 2831	. . . 4 ⊢ {𝑐 ∣ ∃𝑏 ∈ (L‘𝐵)𝑐 = (𝐴 +s 𝑏)} = {𝑧 ∣ ∃𝑚 ∈ (L‘𝐵)𝑧 = (𝐴 +s 𝑚)}
16	8, 15	uneq12i 4113	. . 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 2765	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑎 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑑 +s 𝐵)))
18	17	rexbidv 3187	. . . . . 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 2772	. . . . . . 7 ⊢ (𝑑 = 𝑟 → (𝑤 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑟 +s 𝐵)))
21	20	cbvrexvw 3242	. . . . . 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 2831	. . . 4 ⊢ {𝑎 ∣ ∃𝑑 ∈ (R‘𝐴)𝑎 = (𝑑 +s 𝐵)} = {𝑤 ∣ ∃𝑟 ∈ (R‘𝐴)𝑤 = (𝑟 +s 𝐵)}
24		eqeq1 2765	. . . . . . 7 ⊢ (𝑐 = 𝑡 → (𝑐 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑑)))
25	24	rexbidv 3187	. . . . . 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 2772	. . . . . . 7 ⊢ (𝑑 = 𝑠 → (𝑡 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑠)))
28	27	cbvrexvw 3242	. . . . . 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 2831	. . . 4 ⊢ {𝑐 ∣ ∃𝑑 ∈ (R‘𝐵)𝑐 = (𝐴 +s 𝑑)} = {𝑡 ∣ ∃𝑠 ∈ (R‘𝐵)𝑡 = (𝐴 +s 𝑠)}
31	23, 30	uneq12i 4113	. . 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 2812	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 2145  {cab 2739  ∃wrex 3087   ∪ cun 3897  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   |s ccuts 28145  Lcleft 28211  Rcright 28212   +_s cadds 28345

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec2 28335  df-adds 28346

This theorem is used by:  addsproplem3  28357  leadds1  28375  addsuniflem  28387  addsasslem1  28389  addsasslem2  28390  addbday  28404  addsdilem1  28537