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Theorem addsval2 27900
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.
StepHypRef Expression
1 addsval 27899 . 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 2732 . . . . . . 7 (𝑎 = 𝑦 → (𝑎 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
32rexbidv 3176 . . . . . 6 (𝑎 = 𝑦 → (∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ ( L ‘𝐴)𝑦 = (𝑏 +s 𝐵)))
4 oveq1 7433 . . . . . . . 8 (𝑏 = 𝑙 → (𝑏 +s 𝐵) = (𝑙 +s 𝐵))
54eqeq2d 2739 . . . . . . 7 (𝑏 = 𝑙 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑦 = (𝑙 +s 𝐵)))
65cbvrexvw 3233 . . . . . 6 (∃𝑏 ∈ ( L ‘𝐴)𝑦 = (𝑏 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵))
73, 6bitrdi 286 . . . . 5 (𝑎 = 𝑦 → (∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)))
87cbvabv 2801 . . . 4 {𝑎 ∣ ∃𝑏 ∈ ( L ‘𝐴)𝑎 = (𝑏 +s 𝐵)} = {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑦 = (𝑙 +s 𝐵)}
9 eqeq1 2732 . . . . . . 7 (𝑐 = 𝑧 → (𝑐 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑏)))
109rexbidv 3176 . . . . . 6 (𝑐 = 𝑧 → (∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑏 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑏)))
11 oveq2 7434 . . . . . . . 8 (𝑏 = 𝑚 → (𝐴 +s 𝑏) = (𝐴 +s 𝑚))
1211eqeq2d 2739 . . . . . . 7 (𝑏 = 𝑚 → (𝑧 = (𝐴 +s 𝑏) ↔ 𝑧 = (𝐴 +s 𝑚)))
1312cbvrexvw 3233 . . . . . 6 (∃𝑏 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑏) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚))
1410, 13bitrdi 286 . . . . 5 (𝑐 = 𝑧 → (∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏) ↔ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)))
1514cbvabv 2801 . . . 4 {𝑐 ∣ ∃𝑏 ∈ ( L ‘𝐵)𝑐 = (𝐴 +s 𝑏)} = {𝑧 ∣ ∃𝑚 ∈ ( L ‘𝐵)𝑧 = (𝐴 +s 𝑚)}
168, 15uneq12i 4162 . . 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 2732 . . . . . . 7 (𝑎 = 𝑤 → (𝑎 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑑 +s 𝐵)))
1817rexbidv 3176 . . . . . 6 (𝑎 = 𝑤 → (∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑑 ∈ ( R ‘𝐴)𝑤 = (𝑑 +s 𝐵)))
19 oveq1 7433 . . . . . . . 8 (𝑑 = 𝑟 → (𝑑 +s 𝐵) = (𝑟 +s 𝐵))
2019eqeq2d 2739 . . . . . . 7 (𝑑 = 𝑟 → (𝑤 = (𝑑 +s 𝐵) ↔ 𝑤 = (𝑟 +s 𝐵)))
2120cbvrexvw 3233 . . . . . 6 (∃𝑑 ∈ ( R ‘𝐴)𝑤 = (𝑑 +s 𝐵) ↔ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵))
2218, 21bitrdi 286 . . . . 5 (𝑎 = 𝑤 → (∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵) ↔ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)))
2322cbvabv 2801 . . . 4 {𝑎 ∣ ∃𝑑 ∈ ( R ‘𝐴)𝑎 = (𝑑 +s 𝐵)} = {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑤 = (𝑟 +s 𝐵)}
24 eqeq1 2732 . . . . . . 7 (𝑐 = 𝑡 → (𝑐 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑑)))
2524rexbidv 3176 . . . . . 6 (𝑐 = 𝑡 → (∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑑)))
26 oveq2 7434 . . . . . . . 8 (𝑑 = 𝑠 → (𝐴 +s 𝑑) = (𝐴 +s 𝑠))
2726eqeq2d 2739 . . . . . . 7 (𝑑 = 𝑠 → (𝑡 = (𝐴 +s 𝑑) ↔ 𝑡 = (𝐴 +s 𝑠)))
2827cbvrexvw 3233 . . . . . 6 (∃𝑑 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑑) ↔ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠))
2925, 28bitrdi 286 . . . . 5 (𝑐 = 𝑡 → (∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑) ↔ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)))
3029cbvabv 2801 . . . 4 {𝑐 ∣ ∃𝑑 ∈ ( R ‘𝐵)𝑐 = (𝐴 +s 𝑑)} = {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝐵)𝑡 = (𝐴 +s 𝑠)}
3123, 30uneq12i 4162 . . 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)})
3216, 31oveq12i 7438 . 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)}))
331, 32eqtrdi 2784 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)})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2705  wrex 3067  cun 3947  cfv 6553  (class class class)co 7426   No csur 27593   |s cscut 27735   L cleft 27792   R cright 27793   +s cadds 27896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-1o 8493  df-2o 8494  df-no 27596  df-slt 27597  df-bday 27598  df-sslt 27734  df-scut 27736  df-made 27794  df-old 27795  df-left 27797  df-right 27798  df-norec2 27886  df-adds 27897
This theorem is referenced by:  addsproplem3  27908  sleadd1  27926  addsuniflem  27938  addsasslem1  27940  addsasslem2  27941  addsdilem1  28071
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