Theorem addsunif 28008

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 28007	. 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 7367	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝐵) = (𝑏 +s 𝐵))
7	6	eqeq2d 2748	. . . . . . 7 ⊢ (𝑙 = 𝑏 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
8	7	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑦 = (𝑏 +s 𝐵))
9		eqeq1 2741	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑎 = (𝑏 +s 𝐵)))
10	9	rexbidv 3162	. . . . . 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 2807	. . . 4 ⊢ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} = {𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)}
13		oveq2 7368	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝐴 +s 𝑚) = (𝐴 +s 𝑑))
14	13	eqeq2d 2748	. . . . . . 7 ⊢ (𝑚 = 𝑑 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑧 = (𝐴 +s 𝑑)))
15	14	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑑 ∈ 𝑀 𝑧 = (𝐴 +s 𝑑))
16		eqeq1 2741	. . . . . . 7 ⊢ (𝑧 = 𝑐 → (𝑧 = (𝐴 +s 𝑑) ↔ 𝑐 = (𝐴 +s 𝑑)))
17	16	rexbidv 3162	. . . . . 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 2807	. . . 4 ⊢ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)}
20	12, 19	uneq12i 4107	. . 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 7367	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝐵) = (𝑓 +s 𝐵))
22	21	eqeq2d 2748	. . . . . . 7 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝐵) ↔ 𝑤 = (𝑓 +s 𝐵)))
23	22	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑤 = (𝑓 +s 𝐵))
24		eqeq1 2741	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝐵) ↔ 𝑒 = (𝑓 +s 𝐵)))
25	24	rexbidv 3162	. . . . . 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 2807	. . . 4 ⊢ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} = {𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)}
28		oveq2 7368	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝐴 +s 𝑠) = (𝐴 +s ℎ))
29	28	eqeq2d 2748	. . . . . . 7 ⊢ (𝑠 = ℎ → (𝑡 = (𝐴 +s 𝑠) ↔ 𝑡 = (𝐴 +s ℎ)))
30	29	cbvrexvw 3217	. . . . . 6 ⊢ (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃ℎ ∈ 𝑆 𝑡 = (𝐴 +s ℎ))
31		eqeq1 2741	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝐴 +s ℎ) ↔ 𝑔 = (𝐴 +s ℎ)))
32	31	rexbidv 3162	. . . . . 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 2807	. . . 4 ⊢ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)}
35	27, 34	uneq12i 4107	. . 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 7372	. 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 2790	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 2715  ∃wrex 3062   ∪ cun 3888   class class class wbr 5086  (class class class)co 7360   <<s cslts 27763   |s ccuts 27765   +_s cadds 27965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec2 27955  df-adds 27966

This theorem is referenced by:  addsasslem1  28009  addsasslem2  28010  negsid  28047  addsdilem2  28158  onaddscl  28283  n0cut  28340  twocut  28429  halfcut  28464  pw2cut2  28468  readdscl  28505