Theorem addcuts 28370

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 28369	. 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 7428	. . . . . . . . 9 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝑌) = (𝑏 +s 𝑌))
6	5	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑝 = (𝑙 +s 𝑌) ↔ 𝑝 = (𝑏 +s 𝑌)))
7	6	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑙 ∈ (L‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ (L‘𝑋)𝑝 = (𝑏 +s 𝑌))
8		eqeq1 2764	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑏 +s 𝑌) ↔ 𝑎 = (𝑏 +s 𝑌)))
9	8	rexbidv 3186	. . . . . . 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 2830	. . . . 5 ⊢ {𝑝 ∣ ∃𝑙 ∈ (L‘𝑋)𝑝 = (𝑙 +s 𝑌)} = {𝑎 ∣ ∃𝑏 ∈ (L‘𝑋)𝑎 = (𝑏 +s 𝑌)}
12		oveq2 7429	. . . . . . . . 9 ⊢ (𝑚 = 𝑑 → (𝑋 +s 𝑚) = (𝑋 +s 𝑑))
13	12	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝑞 = (𝑋 +s 𝑚) ↔ 𝑞 = (𝑋 +s 𝑑)))
14	13	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑚 ∈ (L‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ (L‘𝑌)𝑞 = (𝑋 +s 𝑑))
15		eqeq1 2764	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → (𝑞 = (𝑋 +s 𝑑) ↔ 𝑐 = (𝑋 +s 𝑑)))
16	15	rexbidv 3186	. . . . . . 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 2830	. . . . 5 ⊢ {𝑞 ∣ ∃𝑚 ∈ (L‘𝑌)𝑞 = (𝑋 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ (L‘𝑌)𝑐 = (𝑋 +s 𝑑)}
19	11, 18	uneq12i 4165	. . . 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 5109	. . 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 7428	. . . . . . . . 9 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝑌) = (𝑓 +s 𝑌))
22	21	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝑌) ↔ 𝑤 = (𝑓 +s 𝑌)))
23	22	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑟 ∈ (R‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ (R‘𝑋)𝑤 = (𝑓 +s 𝑌))
24		eqeq1 2764	. . . . . . . 8 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝑌) ↔ 𝑒 = (𝑓 +s 𝑌)))
25	24	rexbidv 3186	. . . . . . 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 2830	. . . . 5 ⊢ {𝑤 ∣ ∃𝑟 ∈ (R‘𝑋)𝑤 = (𝑟 +s 𝑌)} = {𝑒 ∣ ∃𝑓 ∈ (R‘𝑋)𝑒 = (𝑓 +s 𝑌)}
28		oveq2 7429	. . . . . . . . 9 ⊢ (𝑠 = ℎ → (𝑋 +s 𝑠) = (𝑋 +s ℎ))
29	28	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝑡 = (𝑋 +s 𝑠) ↔ 𝑡 = (𝑋 +s ℎ)))
30	29	cbvrexvw 3241	. . . . . . 7 ⊢ (∃𝑠 ∈ (R‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ (R‘𝑌)𝑡 = (𝑋 +s ℎ))
31		eqeq1 2764	. . . . . . . 8 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝑋 +s ℎ) ↔ 𝑔 = (𝑋 +s ℎ)))
32	31	rexbidv 3186	. . . . . . 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 2830	. . . . 5 ⊢ {𝑡 ∣ ∃𝑠 ∈ (R‘𝑌)𝑡 = (𝑋 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ (R‘𝑌)𝑔 = (𝑋 +s ℎ)}
35	27, 34	uneq12i 4165	. . . 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 5110	. . 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)})))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ∪ cun 4149  {csn 4584   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Nocsur 28003   <<s cslts 28149  Lcleft 28217  Rcright 28218   +_s cadds 28351

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec2 28341  df-adds 28352

This theorem is used by:  addcuts2  28371  addscld  28372  leadds1  28381  addsuniflem  28393  addsasslem1  28395  addsasslem2  28396