Theorem addsval2 28011

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 28010	. 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 2739	. . . . . . 7 ⊢ (𝑎 = 𝑦 → (𝑎 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
3	2	rexbidv 3177	. . . . . 6 ⊢ (𝑎 = 𝑦 → (∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ ( L ‘𝐴)𝑦 = (𝑏 +s 𝐵)))
4		oveq1 7438	. . . . . . . 8 ⊢ (𝑏 = 𝑙 → (𝑏 +s 𝐵) = (𝑙 +s 𝐵))
5	4	eqeq2d 2746	. . . . . . 7 ⊢ (𝑏 = 𝑙 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑙 +s 𝐵)))
6	5	cbvrexvw 3236	. . . . . 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 2810	. . . 4 ⊢ {𝑎 ∣ ∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵)} = {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)}
9		eqeq1 2739	. . . . . . 7 ⊢ (𝑐 = 𝑧 → (𝑐 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑏)))
10	9	rexbidv 3177	. . . . . 6 ⊢ (𝑐 = 𝑧 → (∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑏 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑏)))
11		oveq2 7439	. . . . . . . 8 ⊢ (𝑏 = 𝑚 → (𝐴 +s 𝑏) = (𝐴 +s 𝑚))
12	11	eqeq2d 2746	. . . . . . 7 ⊢ (𝑏 = 𝑚 → (𝑧 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑚)))
13	12	cbvrexvw 3236	. . . . . 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 2810	. . . 4 ⊢ {𝑐 ∣ ∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏)} = {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}
16	8, 15	uneq12i 4176	. . 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 2739	. . . . . . 7 ⊢ (𝑎 = 𝑤 → (𝑎 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑑 +s 𝐵)))
18	17	rexbidv 3177	. . . . . 6 ⊢ (𝑎 = 𝑤 → (∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑑 ∈ ( R ‘𝐴)𝑤 = (𝑑 +s 𝐵)))
19		oveq1 7438	. . . . . . . 8 ⊢ (𝑑 = 𝑟 → (𝑑 +s 𝐵) = (𝑟 +s 𝐵))
20	19	eqeq2d 2746	. . . . . . 7 ⊢ (𝑑 = 𝑟 → (𝑤 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑟 +s 𝐵)))
21	20	cbvrexvw 3236	. . . . . 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 2810	. . . 4 ⊢ {𝑎 ∣ ∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵)} = {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)}
24		eqeq1 2739	. . . . . . 7 ⊢ (𝑐 = 𝑡 → (𝑐 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑑)))
25	24	rexbidv 3177	. . . . . 6 ⊢ (𝑐 = 𝑡 → (∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑑)))
26		oveq2 7439	. . . . . . . 8 ⊢ (𝑑 = 𝑠 → (𝐴 +s 𝑑) = (𝐴 +s 𝑠))
27	26	eqeq2d 2746	. . . . . . 7 ⊢ (𝑑 = 𝑠 → (𝑡 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑠)))
28	27	cbvrexvw 3236	. . . . . 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 2810	. . . 4 ⊢ {𝑐 ∣ ∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑)} = {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)}
31	23, 30	uneq12i 4176	. . 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 7443	. 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 2791	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 1537   ∈ wcel 2106  {cab 2712  ∃wrex 3068   ∪ cun 3961  ‘cfv 6563  (class class class)co 7431   No csur 27699   |s cscut 27842   L cleft 27899   R cright 27900   +_s cadds 28007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008

This theorem is referenced by:  addsproplem3  28019  sleadd1  28037  addsuniflem  28049  addsasslem1  28051  addsasslem2  28052  addsbday  28065  addsdilem1  28192