Theorem addscllem1 34131

Description: Lemma for addscl (future) Alternate expression for surreal addition. (Contributed by Scott Fenton, 23-Aug-2024.)

Assertion

Ref	Expression
addscllem1	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = ((( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) |s (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵))))))

Proof of Theorem addscllem1

Dummy variables 𝑙 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addsval 34126	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = (({𝑥 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑥 = (𝑙 +s 𝐵)} ∪ {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)}) |s ({𝑥 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑥 = (𝑟 +s 𝐵)} ∪ {𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)})))
2		addsfn 34125	. . . . . . 7 ⊢ +s Fn ( No × No )
3		leftssno 34063	. . . . . . . . 9 ⊢ ( L ‘𝐴) ⊆ No
4	3	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( L ‘𝐴) ⊆ No )
5		snssi 4741	. . . . . . . 8 ⊢ (𝐵 ∈ No → {𝐵} ⊆ No )
6		xpss12 5604	. . . . . . . 8 ⊢ ((( L ‘𝐴) ⊆ No ∧ {𝐵} ⊆ No ) → (( L ‘𝐴) × {𝐵}) ⊆ ( No × No ))
7	4, 5, 6	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( L ‘𝐴) × {𝐵}) ⊆ ( No × No ))
8		ovelimab 7450	. . . . . . 7 ⊢ (( +s Fn ( No × No ) ∧ (( L ‘𝐴) × {𝐵}) ⊆ ( No × No )) → (𝑥 ∈ ( +s “ (( L ‘𝐴) × {𝐵})) ↔ ∃𝑙 ∈ ( L ‘𝐴)∃𝑟 ∈ {𝐵}𝑥 = (𝑙 +s 𝑟)))
9	2, 7, 8	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑥 ∈ ( +s “ (( L ‘𝐴) × {𝐵})) ↔ ∃𝑙 ∈ ( L ‘𝐴)∃𝑟 ∈ {𝐵}𝑥 = (𝑙 +s 𝑟)))
10		oveq2 7283	. . . . . . . . . 10 ⊢ (𝑟 = 𝐵 → (𝑙 +s 𝑟) = (𝑙 +s 𝐵))
11	10	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑟 = 𝐵 → (𝑥 = (𝑙 +s 𝑟) ↔ 𝑥 = (𝑙 +s 𝐵)))
12	11	rexsng 4610	. . . . . . . 8 ⊢ (𝐵 ∈ No → (∃𝑟 ∈ {𝐵}𝑥 = (𝑙 +s 𝑟) ↔ 𝑥 = (𝑙 +s 𝐵)))
13	12	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑟 ∈ {𝐵}𝑥 = (𝑙 +s 𝑟) ↔ 𝑥 = (𝑙 +s 𝐵)))
14	13	rexbidv 3226	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑙 ∈ ( L ‘𝐴)∃𝑟 ∈ {𝐵}𝑥 = (𝑙 +s 𝑟) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑥 = (𝑙 +s 𝐵)))
15	9, 14	bitrd 278	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑥 ∈ ( +s “ (( L ‘𝐴) × {𝐵})) ↔ ∃𝑙 ∈ ( L ‘𝐴)𝑥 = (𝑙 +s 𝐵)))
16	15	abbi2dv 2877	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( +s “ (( L ‘𝐴) × {𝐵})) = {𝑥 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑥 = (𝑙 +s 𝐵)})
17		snssi 4741	. . . . . . . 8 ⊢ (𝐴 ∈ No → {𝐴} ⊆ No )
18		leftssno 34063	. . . . . . . . 9 ⊢ ( L ‘𝐵) ⊆ No
19	18	a1i 11	. . . . . . . 8 ⊢ (𝐵 ∈ No → ( L ‘𝐵) ⊆ No )
20		xpss12 5604	. . . . . . . 8 ⊢ (({𝐴} ⊆ No ∧ ( L ‘𝐵) ⊆ No ) → ({𝐴} × ( L ‘𝐵)) ⊆ ( No × No ))
21	17, 19, 20	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝐴} × ( L ‘𝐵)) ⊆ ( No × No ))
22		ovelimab 7450	. . . . . . 7 ⊢ (( +s Fn ( No × No ) ∧ ({𝐴} × ( L ‘𝐵)) ⊆ ( No × No )) → (𝑦 ∈ ( +s “ ({𝐴} × ( L ‘𝐵))) ↔ ∃𝑟 ∈ {𝐴}∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝑟 +s 𝑙)))
23	2, 21, 22	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑦 ∈ ( +s “ ({𝐴} × ( L ‘𝐵))) ↔ ∃𝑟 ∈ {𝐴}∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝑟 +s 𝑙)))
24		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑟 = 𝐴 → (𝑟 +s 𝑙) = (𝐴 +s 𝑙))
25	24	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑟 = 𝐴 → (𝑦 = (𝑟 +s 𝑙) ↔ 𝑦 = (𝐴 +s 𝑙)))
26	25	rexbidv 3226	. . . . . . . 8 ⊢ (𝑟 = 𝐴 → (∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝑟 +s 𝑙) ↔ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)))
27	26	rexsng 4610	. . . . . . 7 ⊢ (𝐴 ∈ No → (∃𝑟 ∈ {𝐴}∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝑟 +s 𝑙) ↔ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)))
28	27	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑟 ∈ {𝐴}∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝑟 +s 𝑙) ↔ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)))
29	23, 28	bitrd 278	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑦 ∈ ( +s “ ({𝐴} × ( L ‘𝐵))) ↔ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)))
30	29	abbi2dv 2877	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( +s “ ({𝐴} × ( L ‘𝐵))) = {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)})
31	16, 30	uneq12d 4098	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) = ({𝑥 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑥 = (𝑙 +s 𝐵)} ∪ {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)}))
32		rightssno 34064	. . . . . . . . 9 ⊢ ( R ‘𝐴) ⊆ No
33	32	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ No → ( R ‘𝐴) ⊆ No )
34		xpss12 5604	. . . . . . . 8 ⊢ ((( R ‘𝐴) ⊆ No ∧ {𝐵} ⊆ No ) → (( R ‘𝐴) × {𝐵}) ⊆ ( No × No ))
35	33, 5, 34	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( R ‘𝐴) × {𝐵}) ⊆ ( No × No ))
36		ovelimab 7450	. . . . . . 7 ⊢ (( +s Fn ( No × No ) ∧ (( R ‘𝐴) × {𝐵}) ⊆ ( No × No )) → (𝑥 ∈ ( +s “ (( R ‘𝐴) × {𝐵})) ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑙 ∈ {𝐵}𝑥 = (𝑟 +s 𝑙)))
37	2, 35, 36	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑥 ∈ ( +s “ (( R ‘𝐴) × {𝐵})) ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑙 ∈ {𝐵}𝑥 = (𝑟 +s 𝑙)))
38		oveq2 7283	. . . . . . . . . 10 ⊢ (𝑙 = 𝐵 → (𝑟 +s 𝑙) = (𝑟 +s 𝐵))
39	38	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑙 = 𝐵 → (𝑥 = (𝑟 +s 𝑙) ↔ 𝑥 = (𝑟 +s 𝐵)))
40	39	rexsng 4610	. . . . . . . 8 ⊢ (𝐵 ∈ No → (∃𝑙 ∈ {𝐵}𝑥 = (𝑟 +s 𝑙) ↔ 𝑥 = (𝑟 +s 𝐵)))
41	40	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑙 ∈ {𝐵}𝑥 = (𝑟 +s 𝑙) ↔ 𝑥 = (𝑟 +s 𝐵)))
42	41	rexbidv 3226	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑟 ∈ ( R ‘𝐴)∃𝑙 ∈ {𝐵}𝑥 = (𝑟 +s 𝑙) ↔ ∃𝑟 ∈ ( R ‘𝐴)𝑥 = (𝑟 +s 𝐵)))
43	37, 42	bitrd 278	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑥 ∈ ( +s “ (( R ‘𝐴) × {𝐵})) ↔ ∃𝑟 ∈ ( R ‘𝐴)𝑥 = (𝑟 +s 𝐵)))
44	43	abbi2dv 2877	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( +s “ (( R ‘𝐴) × {𝐵})) = {𝑥 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑥 = (𝑟 +s 𝐵)})
45		rightssno 34064	. . . . . . . . 9 ⊢ ( R ‘𝐵) ⊆ No
46	45	a1i 11	. . . . . . . 8 ⊢ (𝐵 ∈ No → ( R ‘𝐵) ⊆ No )
47		xpss12 5604	. . . . . . . 8 ⊢ (({𝐴} ⊆ No ∧ ( R ‘𝐵) ⊆ No ) → ({𝐴} × ( R ‘𝐵)) ⊆ ( No × No ))
48	17, 46, 47	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ({𝐴} × ( R ‘𝐵)) ⊆ ( No × No ))
49		ovelimab 7450	. . . . . . 7 ⊢ (( +s Fn ( No × No ) ∧ ({𝐴} × ( R ‘𝐵)) ⊆ ( No × No )) → (𝑦 ∈ ( +s “ ({𝐴} × ( R ‘𝐵))) ↔ ∃𝑙 ∈ {𝐴}∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝑙 +s 𝑟)))
50	2, 48, 49	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑦 ∈ ( +s “ ({𝐴} × ( R ‘𝐵))) ↔ ∃𝑙 ∈ {𝐴}∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝑙 +s 𝑟)))
51		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑙 = 𝐴 → (𝑙 +s 𝑟) = (𝐴 +s 𝑟))
52	51	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑙 = 𝐴 → (𝑦 = (𝑙 +s 𝑟) ↔ 𝑦 = (𝐴 +s 𝑟)))
53	52	rexbidv 3226	. . . . . . . 8 ⊢ (𝑙 = 𝐴 → (∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝑙 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)))
54	53	rexsng 4610	. . . . . . 7 ⊢ (𝐴 ∈ No → (∃𝑙 ∈ {𝐴}∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝑙 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)))
55	54	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (∃𝑙 ∈ {𝐴}∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝑙 +s 𝑟) ↔ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)))
56	50, 55	bitrd 278	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝑦 ∈ ( +s “ ({𝐴} × ( R ‘𝐵))) ↔ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)))
57	56	abbi2dv 2877	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( +s “ ({𝐴} × ( R ‘𝐵))) = {𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)})
58	44, 57	uneq12d 4098	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵)))) = ({𝑥 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑥 = (𝑟 +s 𝐵)} ∪ {𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)}))
59	31, 58	oveq12d 7293	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ((( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) |s (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵))))) = (({𝑥 ∣ ∃𝑙 ∈ ( L ‘𝐴)𝑥 = (𝑙 +s 𝐵)} ∪ {𝑦 ∣ ∃𝑙 ∈ ( L ‘𝐵)𝑦 = (𝐴 +s 𝑙)}) |s ({𝑥 ∣ ∃𝑟 ∈ ( R ‘𝐴)𝑥 = (𝑟 +s 𝐵)} ∪ {𝑦 ∣ ∃𝑟 ∈ ( R ‘𝐵)𝑦 = (𝐴 +s 𝑟)})))
60	1, 59	eqtr4d 2781	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 +s 𝐵) = ((( +s “ (( L ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( L ‘𝐵)))) |s (( +s “ (( R ‘𝐴) × {𝐵})) ∪ ( +s “ ({𝐴} × ( R ‘𝐵))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065   ∪ cun 3885   ⊆ wss 3887  {csn 4561   × cxp 5587   “ cima 5592   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   No csur 33843   |s cscut 33977   L cleft 34029   R cright 34030   +s cadds 34116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-2o 8298  df-no 33846  df-slt 33847  df-bday 33848  df-sslt 33976  df-scut 33978  df-made 34031  df-old 34032  df-left 34034  df-right 34035  df-norec2 34106  df-adds 34119

This theorem is referenced by: (None)