Theorem addcuts 28149

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
addcuts.1	⊢ (𝜑 → 𝑋 ∈ No )
addcuts.2	⊢ (𝜑 → 𝑌 ∈ No )

Assertion

Ref	Expression
addcuts	⊢ (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 addcuts

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

Step	Hyp	Ref	Expression
1		addcuts.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
2		addcuts.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
3	1, 2	addcutslem 28148	. 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 264	. . 3 ⊢ ((𝑋 +s 𝑌) ∈ No ↔ (𝑋 +s 𝑌) ∈ No )
5		oveq1 7419	. . . . . . . . 9 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝑌) = (𝑏 +s 𝑌))
6	5	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑝 = (𝑙 +s 𝑌) ↔ 𝑝 = (𝑏 +s 𝑌)))
7	6	cbvrexvw 3244	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑝 = (𝑏 +s 𝑌))
8		eqeq1 2767	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑏 +s 𝑌) ↔ 𝑎 = (𝑏 +s 𝑌)))
9	8	rexbidv 3189	. . . . . . 7 ⊢ (𝑝 = 𝑎 → (∃𝑏 ∈ ( L ‘𝑋)𝑝 = (𝑏 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)))
10	7, 9	bitrid 286	. . . . . 6 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)))
11	10	cbvabv 2833	. . . . 5 ⊢ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} = {𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)}
12		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = 𝑑 → (𝑋 +s 𝑚) = (𝑋 +s 𝑑))
13	12	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝑞 = (𝑋 +s 𝑚) ↔ 𝑞 = (𝑋 +s 𝑑)))
14	13	cbvrexvw 3244	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑑))
15		eqeq1 2767	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → (𝑞 = (𝑋 +s 𝑑) ↔ 𝑐 = (𝑋 +s 𝑑)))
16	15	rexbidv 3189	. . . . . . 7 ⊢ (𝑞 = 𝑐 → (∃𝑑 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)))
17	14, 16	bitrid 286	. . . . . 6 ⊢ (𝑞 = 𝑐 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)))
18	17	cbvabv 2833	. . . . 5 ⊢ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}
19	11, 18	uneq12i 4121	. . . 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 5117	. . 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 7419	. . . . . . . . 9 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝑌) = (𝑓 +s 𝑌))
22	21	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝑌) ↔ 𝑤 = (𝑓 +s 𝑌)))
23	22	cbvrexvw 3244	. . . . . . 7 ⊢ (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑤 = (𝑓 +s 𝑌))
24		eqeq1 2767	. . . . . . . 8 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝑌) ↔ 𝑒 = (𝑓 +s 𝑌)))
25	24	rexbidv 3189	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (∃𝑓 ∈ ( R ‘𝑋)𝑤 = (𝑓 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)))
26	23, 25	bitrid 286	. . . . . 6 ⊢ (𝑤 = 𝑒 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)))
27	26	cbvabv 2833	. . . . 5 ⊢ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} = {𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)}
28		oveq2 7420	. . . . . . . . 9 ⊢ (𝑠 = ℎ → (𝑋 +s 𝑠) = (𝑋 +s ℎ))
29	28	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝑡 = (𝑋 +s 𝑠) ↔ 𝑡 = (𝑋 +s ℎ)))
30	29	cbvrexvw 3244	. . . . . . 7 ⊢ (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑡 = (𝑋 +s ℎ))
31		eqeq1 2767	. . . . . . . 8 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝑋 +s ℎ) ↔ 𝑔 = (𝑋 +s ℎ)))
32	31	rexbidv 3189	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (∃ℎ ∈ ( R ‘𝑌)𝑡 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)))
33	30, 32	bitrid 286	. . . . . 6 ⊢ (𝑡 = 𝑔 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)))
34	33	cbvabv 2833	. . . . 5 ⊢ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}
35	27, 34	uneq12i 4121	. . . 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 5118	. . 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 1173	. 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 237	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 1103   = wceq 1570   ∈ wcel 2143  {cab 2741  ∃wrex 3089   ∪ cun 3904  {csn 4590   class class class wbr 5110  ‘cfv 6538  (class class class)co 7412   No csur 27782   <<s cslts 27928   L cleft 27996   R cright 27997   +_s cadds 28130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-slts 27929  df-cuts 27931  df-0s 27978  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec2 28120  df-adds 28131

This theorem is referenced by:  addcuts2  28150  addscld  28151  leadds1  28160  addsuniflem  28172  addsasslem1  28174  addsasslem2  28175