Theorem addsunif 27485

Description: Uniformity theorem for surreal addition. This theorem states that we can use any cuts that define 𝐴 and 𝐵 in the definition of surreal addition. Theorem 3.2 of [Gonshor] p. 15. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsunif.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
addsunif.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
addsunif.3	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
addsunif.4	⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))

Assertion

Ref	Expression
addsunif	⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})))

Distinct variable groups:   𝐴,𝑚   𝐴,𝑠,𝑡   𝑧,𝐴   𝐵,𝑙   𝐵,𝑟,𝑤   𝑦,𝐵   𝐿,𝑙,𝑦   𝑚,𝑀,𝑧   𝑅,𝑟,𝑤   𝑆,𝑠,𝑡

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑤,𝑡,𝑚,𝑠,𝑟,𝑙)   𝐴(𝑦,𝑤,𝑟,𝑙)   𝐵(𝑧,𝑡,𝑚,𝑠)   𝑅(𝑦,𝑧,𝑡,𝑚,𝑠,𝑙)   𝑆(𝑦,𝑧,𝑤,𝑚,𝑟,𝑙)   𝐿(𝑧,𝑤,𝑡,𝑚,𝑠,𝑟)   𝑀(𝑦,𝑤,𝑡,𝑠,𝑟,𝑙)

Proof of Theorem addsunif

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

Step	Hyp	Ref	Expression
1		addsunif.1	. . 3 ⊢ (𝜑 → 𝐿 <<s 𝑅)
2		addsunif.2	. . 3 ⊢ (𝜑 → 𝑀 <<s 𝑆)
3		addsunif.3	. . 3 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
4		addsunif.4	. . 3 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
5	1, 2, 3, 4	addsuniflem 27484	. 2 ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)}) |s ({𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)} ∪ {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)})))
6		oveq1 7416	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝐵) = (𝑏 +s 𝐵))
7	6	eqeq2d 2744	. . . . . . 7 ⊢ (𝑙 = 𝑏 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
8	7	cbvrexvw 3236	. . . . . 6 ⊢ (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑦 = (𝑏 +s 𝐵))
9		eqeq1 2737	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑎 = (𝑏 +s 𝐵)))
10	9	rexbidv 3179	. . . . . 6 ⊢ (𝑦 = 𝑎 → (∃𝑏 ∈ 𝐿 𝑦 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)))
11	8, 10	bitrid 283	. . . . 5 ⊢ (𝑦 = 𝑎 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)))
12	11	cbvabv 2806	. . . 4 ⊢ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} = {𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)}
13		oveq2 7417	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝐴 +s 𝑚) = (𝐴 +s 𝑑))
14	13	eqeq2d 2744	. . . . . . 7 ⊢ (𝑚 = 𝑑 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑧 = (𝐴 +s 𝑑)))
15	14	cbvrexvw 3236	. . . . . 6 ⊢ (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑑 ∈ 𝑀 𝑧 = (𝐴 +s 𝑑))
16		eqeq1 2737	. . . . . . 7 ⊢ (𝑧 = 𝑐 → (𝑧 = (𝐴 +s 𝑑) ↔ 𝑐 = (𝐴 +s 𝑑)))
17	16	rexbidv 3179	. . . . . 6 ⊢ (𝑧 = 𝑐 → (∃𝑑 ∈ 𝑀 𝑧 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)))
18	15, 17	bitrid 283	. . . . 5 ⊢ (𝑧 = 𝑐 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)))
19	18	cbvabv 2806	. . . 4 ⊢ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)}
20	12, 19	uneq12i 4162	. . 3 ⊢ ({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) = ({𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)})
21		oveq1 7416	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝐵) = (𝑓 +s 𝐵))
22	21	eqeq2d 2744	. . . . . . 7 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝐵) ↔ 𝑤 = (𝑓 +s 𝐵)))
23	22	cbvrexvw 3236	. . . . . 6 ⊢ (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑤 = (𝑓 +s 𝐵))
24		eqeq1 2737	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝐵) ↔ 𝑒 = (𝑓 +s 𝐵)))
25	24	rexbidv 3179	. . . . . 6 ⊢ (𝑤 = 𝑒 → (∃𝑓 ∈ 𝑅 𝑤 = (𝑓 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)))
26	23, 25	bitrid 283	. . . . 5 ⊢ (𝑤 = 𝑒 → (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)))
27	26	cbvabv 2806	. . . 4 ⊢ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} = {𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)}
28		oveq2 7417	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝐴 +s 𝑠) = (𝐴 +s ℎ))
29	28	eqeq2d 2744	. . . . . . 7 ⊢ (𝑠 = ℎ → (𝑡 = (𝐴 +s 𝑠) ↔ 𝑡 = (𝐴 +s ℎ)))
30	29	cbvrexvw 3236	. . . . . 6 ⊢ (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃ℎ ∈ 𝑆 𝑡 = (𝐴 +s ℎ))
31		eqeq1 2737	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝐴 +s ℎ) ↔ 𝑔 = (𝐴 +s ℎ)))
32	31	rexbidv 3179	. . . . . 6 ⊢ (𝑡 = 𝑔 → (∃ℎ ∈ 𝑆 𝑡 = (𝐴 +s ℎ) ↔ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)))
33	30, 32	bitrid 283	. . . . 5 ⊢ (𝑡 = 𝑔 → (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)))
34	33	cbvabv 2806	. . . 4 ⊢ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)}
35	27, 34	uneq12i 4162	. . 3 ⊢ ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)}) = ({𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)} ∪ {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)})
36	20, 35	oveq12i 7421	. 2 ⊢ (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})) = (({𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)} ∪ {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)}) |s ({𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)} ∪ {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)}))
37	5, 36	eqtr4di 2791	1 ⊢ (𝜑 → (𝐴 +s 𝐵) = (({𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} ∪ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)}) |s ({𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} ∪ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)})))

Syntax hints:   → wi 4   = wceq 1542  {cab 2710  ∃wrex 3071   ∪ cun 3947   class class class wbr 5149  (class class class)co 7409   <<s csslt 27282   |s cscut 27284   +_s cadds 27443

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-ot 4638  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-2o 8467  df-nadd 8665  df-no 27146  df-slt 27147  df-bday 27148  df-sle 27248  df-sslt 27283  df-scut 27285  df-0s 27325  df-made 27342  df-old 27343  df-left 27345  df-right 27346  df-norec2 27433  df-adds 27444

This theorem is referenced by:  addsasslem1  27486  addsasslem2  27487  negsid  27515  addsdilem2  27607