Theorem addcuts 28048

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 28047	. 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 263	. . 3 ⊢ ((𝑋 +s 𝑌) ∈ No ↔ (𝑋 +s 𝑌) ∈ No )
5		oveq1 7399	. . . . . . . . 9 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝑌) = (𝑏 +s 𝑌))
6	5	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑝 = (𝑙 +s 𝑌) ↔ 𝑝 = (𝑏 +s 𝑌)))
7	6	cbvrexvw 3240	. . . . . . 7 ⊢ (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑝 = (𝑏 +s 𝑌))
8		eqeq1 2765	. . . . . . . 8 ⊢ (𝑝 = 𝑎 → (𝑝 = (𝑏 +s 𝑌) ↔ 𝑎 = (𝑏 +s 𝑌)))
9	8	rexbidv 3185	. . . . . . 7 ⊢ (𝑝 = 𝑎 → (∃𝑏 ∈ ( L ‘𝑋)𝑝 = (𝑏 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)))
10	7, 9	bitrid 285	. . . . . 6 ⊢ (𝑝 = 𝑎 → (∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌) ↔ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)))
11	10	cbvabv 2831	. . . . 5 ⊢ {𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} = {𝑎 ∣ ∃𝑏 ∈ ( L ‘𝑋)𝑎 = (𝑏 +s 𝑌)}
12		oveq2 7400	. . . . . . . . 9 ⊢ (𝑚 = 𝑑 → (𝑋 +s 𝑚) = (𝑋 +s 𝑑))
13	12	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝑞 = (𝑋 +s 𝑚) ↔ 𝑞 = (𝑋 +s 𝑑)))
14	13	cbvrexvw 3240	. . . . . . 7 ⊢ (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑑))
15		eqeq1 2765	. . . . . . . 8 ⊢ (𝑞 = 𝑐 → (𝑞 = (𝑋 +s 𝑑) ↔ 𝑐 = (𝑋 +s 𝑑)))
16	15	rexbidv 3185	. . . . . . 7 ⊢ (𝑞 = 𝑐 → (∃𝑑 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)))
17	14, 16	bitrid 285	. . . . . 6 ⊢ (𝑞 = 𝑐 → (∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚) ↔ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)))
18	17	cbvabv 2831	. . . . 5 ⊢ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ ( L ‘𝑌)𝑐 = (𝑋 +s 𝑑)}
19	11, 18	uneq12i 4119	. . . 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 5106	. . 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 7399	. . . . . . . . 9 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝑌) = (𝑓 +s 𝑌))
22	21	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝑌) ↔ 𝑤 = (𝑓 +s 𝑌)))
23	22	cbvrexvw 3240	. . . . . . 7 ⊢ (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑤 = (𝑓 +s 𝑌))
24		eqeq1 2765	. . . . . . . 8 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝑌) ↔ 𝑒 = (𝑓 +s 𝑌)))
25	24	rexbidv 3185	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (∃𝑓 ∈ ( R ‘𝑋)𝑤 = (𝑓 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)))
26	23, 25	bitrid 285	. . . . . 6 ⊢ (𝑤 = 𝑒 → (∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌) ↔ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)))
27	26	cbvabv 2831	. . . . 5 ⊢ {𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} = {𝑒 ∣ ∃𝑓 ∈ ( R ‘𝑋)𝑒 = (𝑓 +s 𝑌)}
28		oveq2 7400	. . . . . . . . 9 ⊢ (𝑠 = ℎ → (𝑋 +s 𝑠) = (𝑋 +s ℎ))
29	28	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝑡 = (𝑋 +s 𝑠) ↔ 𝑡 = (𝑋 +s ℎ)))
30	29	cbvrexvw 3240	. . . . . . 7 ⊢ (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑡 = (𝑋 +s ℎ))
31		eqeq1 2765	. . . . . . . 8 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝑋 +s ℎ) ↔ 𝑔 = (𝑋 +s ℎ)))
32	31	rexbidv 3185	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (∃ℎ ∈ ( R ‘𝑌)𝑡 = (𝑋 +s ℎ) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)))
33	30, 32	bitrid 285	. . . . . 6 ⊢ (𝑡 = 𝑔 → (∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠) ↔ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)))
34	33	cbvabv 2831	. . . . 5 ⊢ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ ( R ‘𝑌)𝑔 = (𝑋 +s ℎ)}
35	27, 34	uneq12i 4119	. . . 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 5107	. . 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 1167	. 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 236	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 1097   = wceq 1559   ∈ wcel 2141  {cab 2739  ∃wrex 3085   ∪ cun 3902  {csn 4581   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   No csur 27681   <<s cslts 27827   L cleft 27895   R cright 27896   +_s cadds 28029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec2 28019  df-adds 28030

This theorem is referenced by:  addcuts2  28049  addscld  28050  leadds1  28059  addsuniflem  28071  addsasslem1  28073  addsasslem2  28074