Theorem addsunif 28072

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 28071	. 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 7399	. . . . . . . 8 ⊢ (𝑙 = 𝑏 → (𝑙 +s 𝐵) = (𝑏 +s 𝐵))
7	6	eqeq2d 2772	. . . . . . 7 ⊢ (𝑙 = 𝑏 → (𝑦 = (𝑙 +s 𝐵) ↔ 𝑦 = (𝑏 +s 𝐵)))
8	7	cbvrexvw 3240	. . . . . 6 ⊢ (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑦 = (𝑏 +s 𝐵))
9		eqeq1 2765	. . . . . . 7 ⊢ (𝑦 = 𝑎 → (𝑦 = (𝑏 +s 𝐵) ↔ 𝑎 = (𝑏 +s 𝐵)))
10	9	rexbidv 3185	. . . . . 6 ⊢ (𝑦 = 𝑎 → (∃𝑏 ∈ 𝐿 𝑦 = (𝑏 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)))
11	8, 10	bitrid 285	. . . . 5 ⊢ (𝑦 = 𝑎 → (∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵) ↔ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)))
12	11	cbvabv 2831	. . . 4 ⊢ {𝑦 ∣ ∃𝑙 ∈ 𝐿 𝑦 = (𝑙 +s 𝐵)} = {𝑎 ∣ ∃𝑏 ∈ 𝐿 𝑎 = (𝑏 +s 𝐵)}
13		oveq2 7400	. . . . . . . 8 ⊢ (𝑚 = 𝑑 → (𝐴 +s 𝑚) = (𝐴 +s 𝑑))
14	13	eqeq2d 2772	. . . . . . 7 ⊢ (𝑚 = 𝑑 → (𝑧 = (𝐴 +s 𝑚) ↔ 𝑧 = (𝐴 +s 𝑑)))
15	14	cbvrexvw 3240	. . . . . 6 ⊢ (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑑 ∈ 𝑀 𝑧 = (𝐴 +s 𝑑))
16		eqeq1 2765	. . . . . . 7 ⊢ (𝑧 = 𝑐 → (𝑧 = (𝐴 +s 𝑑) ↔ 𝑐 = (𝐴 +s 𝑑)))
17	16	rexbidv 3185	. . . . . 6 ⊢ (𝑧 = 𝑐 → (∃𝑑 ∈ 𝑀 𝑧 = (𝐴 +s 𝑑) ↔ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)))
18	15, 17	bitrid 285	. . . . 5 ⊢ (𝑧 = 𝑐 → (∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚) ↔ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)))
19	18	cbvabv 2831	. . . 4 ⊢ {𝑧 ∣ ∃𝑚 ∈ 𝑀 𝑧 = (𝐴 +s 𝑚)} = {𝑐 ∣ ∃𝑑 ∈ 𝑀 𝑐 = (𝐴 +s 𝑑)}
20	12, 19	uneq12i 4119	. . 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 7399	. . . . . . . 8 ⊢ (𝑟 = 𝑓 → (𝑟 +s 𝐵) = (𝑓 +s 𝐵))
22	21	eqeq2d 2772	. . . . . . 7 ⊢ (𝑟 = 𝑓 → (𝑤 = (𝑟 +s 𝐵) ↔ 𝑤 = (𝑓 +s 𝐵)))
23	22	cbvrexvw 3240	. . . . . 6 ⊢ (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑤 = (𝑓 +s 𝐵))
24		eqeq1 2765	. . . . . . 7 ⊢ (𝑤 = 𝑒 → (𝑤 = (𝑓 +s 𝐵) ↔ 𝑒 = (𝑓 +s 𝐵)))
25	24	rexbidv 3185	. . . . . 6 ⊢ (𝑤 = 𝑒 → (∃𝑓 ∈ 𝑅 𝑤 = (𝑓 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)))
26	23, 25	bitrid 285	. . . . 5 ⊢ (𝑤 = 𝑒 → (∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵) ↔ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)))
27	26	cbvabv 2831	. . . 4 ⊢ {𝑤 ∣ ∃𝑟 ∈ 𝑅 𝑤 = (𝑟 +s 𝐵)} = {𝑒 ∣ ∃𝑓 ∈ 𝑅 𝑒 = (𝑓 +s 𝐵)}
28		oveq2 7400	. . . . . . . 8 ⊢ (𝑠 = ℎ → (𝐴 +s 𝑠) = (𝐴 +s ℎ))
29	28	eqeq2d 2772	. . . . . . 7 ⊢ (𝑠 = ℎ → (𝑡 = (𝐴 +s 𝑠) ↔ 𝑡 = (𝐴 +s ℎ)))
30	29	cbvrexvw 3240	. . . . . 6 ⊢ (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃ℎ ∈ 𝑆 𝑡 = (𝐴 +s ℎ))
31		eqeq1 2765	. . . . . . 7 ⊢ (𝑡 = 𝑔 → (𝑡 = (𝐴 +s ℎ) ↔ 𝑔 = (𝐴 +s ℎ)))
32	31	rexbidv 3185	. . . . . 6 ⊢ (𝑡 = 𝑔 → (∃ℎ ∈ 𝑆 𝑡 = (𝐴 +s ℎ) ↔ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)))
33	30, 32	bitrid 285	. . . . 5 ⊢ (𝑡 = 𝑔 → (∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠) ↔ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)))
34	33	cbvabv 2831	. . . 4 ⊢ {𝑡 ∣ ∃𝑠 ∈ 𝑆 𝑡 = (𝐴 +s 𝑠)} = {𝑔 ∣ ∃ℎ ∈ 𝑆 𝑔 = (𝐴 +s ℎ)}
35	27, 34	uneq12i 4119	. . 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 7404	. 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 2814	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 1559  {cab 2739  ∃wrex 3085   ∪ cun 3902   class class class wbr 5099  (class class class)co 7392   <<s cslts 27827   |s ccuts 27829   +_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-ot 4590  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-les 27786  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:  addsasslem1  28073  addsasslem2  28074  negsid  28111  addsdilem2  28222  onaddscl  28347  n0cut  28404  twocut  28493  halfcut  28528  pw2cut2  28532  readdscl  28569