Theorem addsunif 28050

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 28049	. 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 7438	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝐵) = (𝑏 +s 𝐵))
7	6	eqeq2d 2746	. . . . . . 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 2739	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑎 = (𝑏 +s 𝐵)))
10	9	rexbidv 3177	. . . . . 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 2810	. . . 4 ⊢ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} = {𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)}
13		oveq2 7439	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝐴 +s 𝑚) = (𝐴 +s 𝑑))
14	13	eqeq2d 2746	. . . . . . 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 2739	. . . . . . 7 ⊢ (𝑧 = 𝑐 → (𝑧 = (𝐴 +s 𝑑) ↔ 𝑐 = (𝐴 +s 𝑑)))
17	16	rexbidv 3177	. . . . . 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 2810	. . . 4 ⊢ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)}
20	12, 19	uneq12i 4176	. . 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 7438	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝐵) = (𝑓 +s 𝐵))
22	21	eqeq2d 2746	. . . . . . 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 2739	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝐵) ↔ 𝑒 = (𝑓 +s 𝐵)))
25	24	rexbidv 3177	. . . . . 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 2810	. . . 4 ⊢ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} = {𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)}
28		oveq2 7439	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝐴 +s 𝑠) = (𝐴 +s ℎ))
29	28	eqeq2d 2746	. . . . . . 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 2739	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝐴 +s ℎ) ↔ 𝑔 = (𝐴 +s ℎ)))
32	31	rexbidv 3177	. . . . . 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 2810	. . . 4 ⊢ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)}
35	27, 34	uneq12i 4176	. . 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 7443	. 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 2793	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 1537  {cab 2712  ∃wrex 3068   ∪ cun 3961   class class class wbr 5148  (class class class)co 7431   <<s csslt 27840   |s cscut 27842   +_s cadds 28007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008

This theorem is referenced by:  addsasslem1  28051  addsasslem2  28052  negsid  28088  addsdilem2  28193  onaddscl  28301  n0scut  28353  1p1e2s  28415  nohalf  28422  halfcut  28431  readdscl  28446