Metamath Proof Explorer


Theorem noinfres

Description: The restriction of surreal infimum when there is no minimum. (Contributed by Scott Fenton, 8-Aug-2024)

Ref Expression
Hypothesis noinfres.1 T = if x B y B ¬ y < s x ι x B | y B ¬ y < s x dom ι x B | y B ¬ y < s x 1 𝑜 g y | u B y dom u v B ¬ u < s v u suc y = v suc y ι x | u B g dom u v B ¬ u < s v u suc g = v suc g u g = x
Assertion noinfres ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G T suc G = U suc G

Proof

Step Hyp Ref Expression
1 noinfres.1 T = if x B y B ¬ y < s x ι x B | y B ¬ y < s x dom ι x B | y B ¬ y < s x 1 𝑜 g y | u B y dom u v B ¬ u < s v u suc y = v suc y ι x | u B g dom u v B ¬ u < s v u suc g = v suc g u g = x
2 dmres dom T suc G = suc G dom T
3 1 noinfno B No B V T No
4 3 3ad2ant2 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G T No
5 nodmord T No Ord dom T
6 4 5 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G Ord dom T
7 simp31 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G U B
8 simp32 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G G dom U
9 simp33 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G v B ¬ U < s v U suc G = v suc G
10 dmeq b = U dom b = dom U
11 10 eleq2d b = U G dom b G dom U
12 breq1 b = U b < s c U < s c
13 12 notbid b = U ¬ b < s c ¬ U < s c
14 reseq1 b = U b suc G = U suc G
15 14 eqeq1d b = U b suc G = c suc G U suc G = c suc G
16 13 15 imbi12d b = U ¬ b < s c b suc G = c suc G ¬ U < s c U suc G = c suc G
17 16 ralbidv b = U c B ¬ b < s c b suc G = c suc G c B ¬ U < s c U suc G = c suc G
18 breq2 c = v U < s c U < s v
19 18 notbid c = v ¬ U < s c ¬ U < s v
20 reseq1 c = v c suc G = v suc G
21 20 eqeq2d c = v U suc G = c suc G U suc G = v suc G
22 19 21 imbi12d c = v ¬ U < s c U suc G = c suc G ¬ U < s v U suc G = v suc G
23 22 cbvralvw c B ¬ U < s c U suc G = c suc G v B ¬ U < s v U suc G = v suc G
24 17 23 bitrdi b = U c B ¬ b < s c b suc G = c suc G v B ¬ U < s v U suc G = v suc G
25 11 24 anbi12d b = U G dom b c B ¬ b < s c b suc G = c suc G G dom U v B ¬ U < s v U suc G = v suc G
26 25 rspcev U B G dom U v B ¬ U < s v U suc G = v suc G b B G dom b c B ¬ b < s c b suc G = c suc G
27 7 8 9 26 syl12anc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G b B G dom b c B ¬ b < s c b suc G = c suc G
28 eleq1 a = G a dom b G dom b
29 suceq a = G suc a = suc G
30 29 reseq2d a = G b suc a = b suc G
31 29 reseq2d a = G c suc a = c suc G
32 30 31 eqeq12d a = G b suc a = c suc a b suc G = c suc G
33 32 imbi2d a = G ¬ b < s c b suc a = c suc a ¬ b < s c b suc G = c suc G
34 33 ralbidv a = G c B ¬ b < s c b suc a = c suc a c B ¬ b < s c b suc G = c suc G
35 28 34 anbi12d a = G a dom b c B ¬ b < s c b suc a = c suc a G dom b c B ¬ b < s c b suc G = c suc G
36 35 rexbidv a = G b B a dom b c B ¬ b < s c b suc a = c suc a b B G dom b c B ¬ b < s c b suc G = c suc G
37 36 elabg G dom U G a | b B a dom b c B ¬ b < s c b suc a = c suc a b B G dom b c B ¬ b < s c b suc G = c suc G
38 8 37 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G G a | b B a dom b c B ¬ b < s c b suc a = c suc a b B G dom b c B ¬ b < s c b suc G = c suc G
39 27 38 mpbird ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G G a | b B a dom b c B ¬ b < s c b suc a = c suc a
40 1 noinfdm ¬ x B y B ¬ y < s x dom T = a | b B a dom b c B ¬ b < s c b suc a = c suc a
41 40 3ad2ant1 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G dom T = a | b B a dom b c B ¬ b < s c b suc a = c suc a
42 39 41 eleqtrrd ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G G dom T
43 ordsucss Ord dom T G dom T suc G dom T
44 6 42 43 sylc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G suc G dom T
45 df-ss suc G dom T suc G dom T = suc G
46 44 45 sylib ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G suc G dom T = suc G
47 2 46 syl5eq ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G dom T suc G = suc G
48 dmres dom U suc G = suc G dom U
49 simp2l ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G B No
50 49 7 sseldd ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G U No
51 nodmon U No dom U On
52 50 51 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G dom U On
53 eloni dom U On Ord dom U
54 52 53 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G Ord dom U
55 ordsucss Ord dom U G dom U suc G dom U
56 54 8 55 sylc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G suc G dom U
57 df-ss suc G dom U suc G dom U = suc G
58 56 57 sylib ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G suc G dom U = suc G
59 48 58 syl5eq ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G dom U suc G = suc G
60 47 59 eqtr4d ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G dom T suc G = dom U suc G
61 47 eleq2d ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a dom T suc G a suc G
62 simpl1 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G ¬ x B y B ¬ y < s x
63 simpl2 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G B No B V
64 simpl31 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G U B
65 56 sselda ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G a dom U
66 50 adantr ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G U No
67 66 51 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G dom U On
68 simpl32 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G G dom U
69 onelon dom U On G dom U G On
70 67 68 69 syl2anc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G G On
71 sucelon G On suc G On
72 70 71 sylib ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G suc G On
73 eloni suc G On Ord suc G
74 72 73 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G Ord suc G
75 simpr ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G a suc G
76 ordsucss Ord suc G a suc G suc a suc G
77 74 75 76 sylc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G suc a suc G
78 simpl33 ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G v B ¬ U < s v U suc G = v suc G
79 reseq1 U suc G = v suc G U suc G suc a = v suc G suc a
80 resabs1 suc a suc G U suc G suc a = U suc a
81 resabs1 suc a suc G v suc G suc a = v suc a
82 80 81 eqeq12d suc a suc G U suc G suc a = v suc G suc a U suc a = v suc a
83 79 82 syl5ib suc a suc G U suc G = v suc G U suc a = v suc a
84 83 imim2d suc a suc G ¬ U < s v U suc G = v suc G ¬ U < s v U suc a = v suc a
85 84 ralimdv suc a suc G v B ¬ U < s v U suc G = v suc G v B ¬ U < s v U suc a = v suc a
86 77 78 85 sylc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G v B ¬ U < s v U suc a = v suc a
87 1 noinffv ¬ x B y B ¬ y < s x B No B V U B a dom U v B ¬ U < s v U suc a = v suc a T a = U a
88 62 63 64 65 86 87 syl113anc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G T a = U a
89 75 fvresd ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G T suc G a = T a
90 75 fvresd ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G U suc G a = U a
91 88 89 90 3eqtr4d ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G T suc G a = U suc G a
92 91 ex ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a suc G T suc G a = U suc G a
93 61 92 sylbid ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a dom T suc G T suc G a = U suc G a
94 93 ralrimiv ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G a dom T suc G T suc G a = U suc G a
95 nofun T No Fun T
96 95 funresd T No Fun T suc G
97 4 96 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G Fun T suc G
98 nofun U No Fun U
99 98 funresd U No Fun U suc G
100 50 99 syl ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G Fun U suc G
101 eqfunfv Fun T suc G Fun U suc G T suc G = U suc G dom T suc G = dom U suc G a dom T suc G T suc G a = U suc G a
102 97 100 101 syl2anc ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G T suc G = U suc G dom T suc G = dom U suc G a dom T suc G T suc G a = U suc G a
103 60 94 102 mpbir2and ¬ x B y B ¬ y < s x B No B V U B G dom U v B ¬ U < s v U suc G = v suc G T suc G = U suc G