Theorem addscut 28020

Description: Demonstrate the cut properties of surreal addition. This gives us closure together with a pair of set-less-than relationships for surreal addition. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addscut.1	⊢ (𝜑 → 𝑋 ∈ No )
addscut.2	⊢ (𝜑 → 𝑌 ∈ No )

Assertion

Ref	Expression
addscut	⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))

Distinct variable groups:   𝑋,𝑝,𝑙   𝑋,𝑞,𝑚   𝑤,𝑋,𝑟   𝑡,𝑋,𝑠   𝑌,𝑝,𝑙   𝑌,𝑞,𝑚   𝑤,𝑌,𝑟   𝑡,𝑌,𝑠

Allowed substitution hints:   𝜑(𝑤,𝑡,𝑚,𝑠,𝑟,𝑞,𝑝,𝑙)

Proof of Theorem addscut

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addscut.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
2		addscut.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
3	1, 2	addscutlem 28019	. 2 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})))
4		biid 261	. . 3 ⊢ ((𝑋 +s 𝑌) ∈ No ↔ (𝑋 +s 𝑌) ∈ No )
5		oveq1 7452	. . . . . . . . 9 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝑌) = (𝑏 +s 𝑌))
6	5	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑝 = (𝑙 +s 𝑌) ↔ 𝑝 = (𝑏 +s 𝑌)))
7	6	cbvrexvw 3239	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑝 = (𝑏 +s 𝑌))
8		eqeq1 2738	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑏 +s 𝑌) ↔ 𝑎 = (𝑏 +s 𝑌)))
9	8	rexbidv 3181	. . . . . . 7 ⊢ (𝑝 = 𝑎 → (∃𝑏 ∈ ( L ‘𝑋)𝑝 = (𝑏 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)))
10	7, 9	bitrid 283	. . . . . 6 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)))
11	10	cbvabv 2809	. . . . 5 ⊢ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} = {𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)}
12		oveq2 7453	. . . . . . . . 9 ⊢ (𝑚 = 𝑑 → (𝑋 +s 𝑚) = (𝑋 +s 𝑑))
13	12	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝑞 = (𝑋 +s 𝑚) ↔ 𝑞 = (𝑋 +s 𝑑)))
14	13	cbvrexvw 3239	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑑))
15		eqeq1 2738	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → (𝑞 = (𝑋 +s 𝑑) ↔ 𝑐 = (𝑋 +s 𝑑)))
16	15	rexbidv 3181	. . . . . . 7 ⊢ (𝑞 = 𝑐 → (∃𝑑 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)))
17	14, 16	bitrid 283	. . . . . 6 ⊢ (𝑞 = 𝑐 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)))
18	17	cbvabv 2809	. . . . 5 ⊢ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}
19	11, 18	uneq12i 4183	. . . 4 ⊢ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) = ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)})
20	19	breq1i 5176	. . 3 ⊢ (({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ↔ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)})
21		oveq1 7452	. . . . . . . . 9 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝑌) = (𝑓 +s 𝑌))
22	21	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝑌) ↔ 𝑤 = (𝑓 +s 𝑌)))
23	22	cbvrexvw 3239	. . . . . . 7 ⊢ (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑤 = (𝑓 +s 𝑌))
24		eqeq1 2738	. . . . . . . 8 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝑌) ↔ 𝑒 = (𝑓 +s 𝑌)))
25	24	rexbidv 3181	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (∃𝑓 ∈ ( R ‘𝑋)𝑤 = (𝑓 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)))
26	23, 25	bitrid 283	. . . . . 6 ⊢ (𝑤 = 𝑒 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)))
27	26	cbvabv 2809	. . . . 5 ⊢ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} = {𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)}
28		oveq2 7453	. . . . . . . . 9 ⊢ (𝑠 = ℎ → (𝑋 +s 𝑠) = (𝑋 +s ℎ))
29	28	eqeq2d 2745	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝑡 = (𝑋 +s 𝑠) ↔ 𝑡 = (𝑋 +s ℎ)))
30	29	cbvrexvw 3239	. . . . . . 7 ⊢ (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑡 = (𝑋 +s ℎ))
31		eqeq1 2738	. . . . . . . 8 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝑋 +s ℎ) ↔ 𝑔 = (𝑋 +s ℎ)))
32	31	rexbidv 3181	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (∃ℎ ∈ ( R ‘𝑌)𝑡 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)))
33	30, 32	bitrid 283	. . . . . 6 ⊢ (𝑡 = 𝑔 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)))
34	33	cbvabv 2809	. . . . 5 ⊢ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}
35	27, 34	uneq12i 4183	. . . 4 ⊢ ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) = ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})
36	35	breq2i 5177	. . 3 ⊢ ({(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)}) ↔ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}))
37	4, 20, 36	3anbi123i 1155	. 2 ⊢ (((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})) ↔ ((𝑋 +s 𝑌) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)} ∪ {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)})))
38	3, 37	sylibr 234	1 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1537   ∈ wcel 2103  {cab 2711  ∃wrex 3072   ∪ cun 3968  {csn 4648   class class class wbr 5169  ‘cfv 6572  (class class class)co 7445   No csur 27693   <<s csslt 27834   L cleft 27893   R cright 27894   +_s cadds 28001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-se 5655  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-1o 8518  df-2o 8519  df-nadd 8718  df-no 27696  df-slt 27697  df-bday 27698  df-sslt 27835  df-scut 27837  df-0s 27878  df-made 27895  df-old 27896  df-left 27898  df-right 27899  df-norec2 27991  df-adds 28002

This theorem is referenced by:  addscut2  28021  addscld  28022  sleadd1  28031  addsuniflem  28043  addsasslem1  28045  addsasslem2  28046