Metamath Proof Explorer


Theorem noinfbnd1lem3

Description: Lemma for noinfbnd1 . If U is a prolongment of T and in B , then ( Udom T ) is not 1o . (Contributed by Scott Fenton, 9-Aug-2024)

Ref Expression
Hypothesis noinfbnd1.1 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 noinfbnd1lem3 ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T 1 𝑜

Proof

Step Hyp Ref Expression
1 noinfbnd1.1 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 1 noinfno B No B V T No
3 2 3ad2ant2 ¬ x B y B ¬ y < s x B No B V U B U dom T = T T No
4 nodmord T No Ord dom T
5 ordirr Ord dom T ¬ dom T dom T
6 3 4 5 3syl ¬ x B y B ¬ y < s x B No B V U B U dom T = T ¬ dom T dom T
7 simpl3l ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 U B
8 ndmfv ¬ dom T dom U U dom T =
9 1n0 1 𝑜
10 9 necomi 1 𝑜
11 neeq1 U dom T = U dom T 1 𝑜 1 𝑜
12 10 11 mpbiri U dom T = U dom T 1 𝑜
13 12 neneqd U dom T = ¬ U dom T = 1 𝑜
14 8 13 syl ¬ dom T dom U ¬ U dom T = 1 𝑜
15 14 con4i U dom T = 1 𝑜 dom T dom U
16 15 adantl ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 dom T dom U
17 simpl2l ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 B No
18 17 7 sseldd ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 U No
19 18 adantr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U No
20 17 adantr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q B No
21 simprl ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q q B
22 20 21 sseldd ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q q No
23 3 adantr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 T No
24 nodmon T No dom T On
25 23 24 syl ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 dom T On
26 25 adantr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q dom T On
27 simpl3r ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 U dom T = T
28 27 adantr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U dom T = T
29 simpll1 ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q ¬ x B y B ¬ y < s x
30 simpll2 ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q B No B V
31 simpll3 ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U B U dom T = T
32 simpr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q q B ¬ U < s q
33 1 noinfbnd1lem2 ¬ x B y B ¬ y < s x B No B V U B U dom T = T q B ¬ U < s q q dom T = T
34 29 30 31 32 33 syl112anc ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q q dom T = T
35 28 34 eqtr4d ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U dom T = q dom T
36 simplr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U dom T = 1 𝑜
37 simprr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q ¬ U < s q
38 nogesgn1ores U No q No dom T On U dom T = q dom T U dom T = 1 𝑜 ¬ U < s q U suc dom T = q suc dom T
39 19 22 26 35 36 37 38 syl321anc ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U suc dom T = q suc dom T
40 39 expr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U suc dom T = q suc dom T
41 40 ralrimiva ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 q B ¬ U < s q U suc dom T = q suc dom T
42 dmeq p = U dom p = dom U
43 42 eleq2d p = U dom T dom p dom T dom U
44 breq1 p = U p < s q U < s q
45 44 notbid p = U ¬ p < s q ¬ U < s q
46 reseq1 p = U p suc dom T = U suc dom T
47 46 eqeq1d p = U p suc dom T = q suc dom T U suc dom T = q suc dom T
48 45 47 imbi12d p = U ¬ p < s q p suc dom T = q suc dom T ¬ U < s q U suc dom T = q suc dom T
49 48 ralbidv p = U q B ¬ p < s q p suc dom T = q suc dom T q B ¬ U < s q U suc dom T = q suc dom T
50 43 49 anbi12d p = U dom T dom p q B ¬ p < s q p suc dom T = q suc dom T dom T dom U q B ¬ U < s q U suc dom T = q suc dom T
51 50 rspcev U B dom T dom U q B ¬ U < s q U suc dom T = q suc dom T p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
52 7 16 41 51 syl12anc ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
53 1 noinfdm ¬ x B y B ¬ y < s x dom T = z | p B z dom p q B ¬ p < s q p suc z = q suc z
54 53 eleq2d ¬ x B y B ¬ y < s x dom T dom T dom T z | p B z dom p q B ¬ p < s q p suc z = q suc z
55 54 3ad2ant1 ¬ x B y B ¬ y < s x B No B V U B U dom T = T dom T dom T dom T z | p B z dom p q B ¬ p < s q p suc z = q suc z
56 eleq1 z = dom T z dom p dom T dom p
57 suceq z = dom T suc z = suc dom T
58 57 reseq2d z = dom T p suc z = p suc dom T
59 57 reseq2d z = dom T q suc z = q suc dom T
60 58 59 eqeq12d z = dom T p suc z = q suc z p suc dom T = q suc dom T
61 60 imbi2d z = dom T ¬ p < s q p suc z = q suc z ¬ p < s q p suc dom T = q suc dom T
62 61 ralbidv z = dom T q B ¬ p < s q p suc z = q suc z q B ¬ p < s q p suc dom T = q suc dom T
63 56 62 anbi12d z = dom T z dom p q B ¬ p < s q p suc z = q suc z dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
64 63 rexbidv z = dom T p B z dom p q B ¬ p < s q p suc z = q suc z p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
65 64 elabg dom T On dom T z | p B z dom p q B ¬ p < s q p suc z = q suc z p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
66 3 24 65 3syl ¬ x B y B ¬ y < s x B No B V U B U dom T = T dom T z | p B z dom p q B ¬ p < s q p suc z = q suc z p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
67 55 66 bitrd ¬ x B y B ¬ y < s x B No B V U B U dom T = T dom T dom T p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
68 67 adantr ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 dom T dom T p B dom T dom p q B ¬ p < s q p suc dom T = q suc dom T
69 52 68 mpbird ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T = 1 𝑜 dom T dom T
70 6 69 mtand ¬ x B y B ¬ y < s x B No B V U B U dom T = T ¬ U dom T = 1 𝑜
71 70 neqned ¬ x B y B ¬ y < s x B No B V U B U dom T = T U dom T 1 𝑜