Metamath Proof Explorer


Theorem weiunpo

Description: A partial ordering on an indexed union can be constructed from a well-ordering on its index class and a collection of partial orderings on its members. (Contributed by Matthew House, 23-Aug-2025)

Ref Expression
Hypotheses weiun.1 F = w x A B ι u x A | w B | v x A | w B ¬ v R u
weiun.2 T = y z | y x A B z x A B F y R F z F y = F z y F y / x S z
Assertion weiunpo R We A R Se A x A S Po B T Po x A B

Proof

Step Hyp Ref Expression
1 weiun.1 F = w x A B ι u x A | w B | v x A | w B ¬ v R u
2 weiun.2 T = y z | y x A B z x A B F y R F z F y = F z y F y / x S z
3 simpl1 R We A R Se A x A S Po B p x A B q x A B r x A B R We A
4 weso R We A R Or A
5 3 4 syl R We A R Se A x A S Po B p x A B q x A B r x A B R Or A
6 simpl2 R We A R Se A x A S Po B p x A B q x A B r x A B R Se A
7 1 2 3 6 weiunlem2 R We A R Se A x A S Po B p x A B q x A B r x A B F : x A B A t x A B t F t / x B s A t s / x B ¬ s R F t
8 7 simp1d R We A R Se A x A S Po B p x A B q x A B r x A B F : x A B A
9 simpr1 R We A R Se A x A S Po B p x A B q x A B r x A B p x A B
10 8 9 ffvelcdmd R We A R Se A x A S Po B p x A B q x A B r x A B F p A
11 sonr R Or A F p A ¬ F p R F p
12 5 10 11 syl2anc R We A R Se A x A S Po B p x A B q x A B r x A B ¬ F p R F p
13 csbeq1 s = F p s / x S = F p / x S
14 csbeq1 s = F p s / x B = F p / x B
15 13 14 poeq12d s = F p s / x S Po s / x B F p / x S Po F p / x B
16 simpl3 R We A R Se A x A S Po B p x A B q x A B r x A B x A S Po B
17 nfv s S Po B
18 nfcsb1v _ x s / x S
19 nfcsb1v _ x s / x B
20 18 19 nfpo x s / x S Po s / x B
21 csbeq1a x = s S = s / x S
22 csbeq1a x = s B = s / x B
23 21 22 poeq12d x = s S Po B s / x S Po s / x B
24 17 20 23 cbvralw x A S Po B s A s / x S Po s / x B
25 16 24 sylib R We A R Se A x A S Po B p x A B q x A B r x A B s A s / x S Po s / x B
26 15 25 10 rspcdva R We A R Se A x A S Po B p x A B q x A B r x A B F p / x S Po F p / x B
27 id t = p t = p
28 fveq2 t = p F t = F p
29 28 csbeq1d t = p F t / x B = F p / x B
30 27 29 eleq12d t = p t F t / x B p F p / x B
31 7 simp2d R We A R Se A x A S Po B p x A B q x A B r x A B t x A B t F t / x B
32 30 31 9 rspcdva R We A R Se A x A S Po B p x A B q x A B r x A B p F p / x B
33 poirr F p / x S Po F p / x B p F p / x B ¬ p F p / x S p
34 26 32 33 syl2anc R We A R Se A x A S Po B p x A B q x A B r x A B ¬ p F p / x S p
35 34 intnand R We A R Se A x A S Po B p x A B q x A B r x A B ¬ F p = F p p F p / x S p
36 ioran ¬ F p R F p F p = F p p F p / x S p ¬ F p R F p ¬ F p = F p p F p / x S p
37 12 35 36 sylanbrc R We A R Se A x A S Po B p x A B q x A B r x A B ¬ F p R F p F p = F p p F p / x S p
38 1 2 weiunlem1 p T p p x A B p x A B F p R F p F p = F p p F p / x S p
39 38 simprbi p T p F p R F p F p = F p p F p / x S p
40 37 39 nsyl R We A R Se A x A S Po B p x A B q x A B r x A B ¬ p T p
41 simpr3 R We A R Se A x A S Po B p x A B q x A B r x A B r x A B
42 9 41 jca R We A R Se A x A S Po B p x A B q x A B r x A B p x A B r x A B
43 1 2 weiunlem1 p T q p x A B q x A B F p R F q F p = F q p F p / x S q
44 43 simprbi p T q F p R F q F p = F q p F p / x S q
45 1 2 weiunlem1 q T r q x A B r x A B F q R F r F q = F r q F q / x S r
46 45 simprbi q T r F q R F r F q = F r q F q / x S r
47 simpr2 R We A R Se A x A S Po B p x A B q x A B r x A B q x A B
48 8 47 ffvelcdmd R We A R Se A x A S Po B p x A B q x A B r x A B F q A
49 8 41 ffvelcdmd R We A R Se A x A S Po B p x A B q x A B r x A B F r A
50 sotr R Or A F p A F q A F r A F p R F q F q R F r F p R F r
51 5 10 48 49 50 syl13anc R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q R F r F p R F r
52 orc F p R F r F p R F r F p = F r p F p / x S r
53 51 52 syl6 R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q R F r F p R F r F p = F r p F p / x S r
54 simprll R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q R F r F p = F q
55 simprr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q R F r F q R F r
56 54 55 eqbrtrd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q R F r F p R F r
57 56 orcd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q R F r F p R F r F p = F r p F p / x S r
58 57 ex R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q R F r F p R F r F p = F r p F p / x S r
59 simprl R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q = F r q F q / x S r F p R F q
60 simprrl R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q = F r q F q / x S r F q = F r
61 59 60 breqtrd R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q = F r q F q / x S r F p R F r
62 61 orcd R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q = F r q F q / x S r F p R F r F p = F r p F p / x S r
63 62 ex R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F q = F r q F q / x S r F p R F r F p = F r p F p / x S r
64 simprll R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p = F q
65 simprrl R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F q = F r
66 64 65 eqtrd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p = F r
67 simprlr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r p F p / x S q
68 simprrr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r q F q / x S r
69 64 csbeq1d R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p / x S = F q / x S
70 69 breqd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r q F p / x S r q F q / x S r
71 68 70 mpbird R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r q F p / x S r
72 26 adantr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p / x S Po F p / x B
73 32 adantr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r p F p / x B
74 id t = q t = q
75 fveq2 t = q F t = F q
76 75 csbeq1d t = q F t / x B = F q / x B
77 74 76 eleq12d t = q t F t / x B q F q / x B
78 77 31 47 rspcdva R We A R Se A x A S Po B p x A B q x A B r x A B q F q / x B
79 78 adantr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r q F q / x B
80 64 csbeq1d R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p / x B = F q / x B
81 79 80 eleqtrrd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r q F p / x B
82 id t = r t = r
83 fveq2 t = r F t = F r
84 83 csbeq1d t = r F t / x B = F r / x B
85 82 84 eleq12d t = r t F t / x B r F r / x B
86 85 31 41 rspcdva R We A R Se A x A S Po B p x A B q x A B r x A B r F r / x B
87 86 adantr R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r r F r / x B
88 66 csbeq1d R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p / x B = F r / x B
89 87 88 eleqtrrd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r r F p / x B
90 potr F p / x S Po F p / x B p F p / x B q F p / x B r F p / x B p F p / x S q q F p / x S r p F p / x S r
91 72 73 81 89 90 syl13anc R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r p F p / x S q q F p / x S r p F p / x S r
92 67 71 91 mp2and R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r p F p / x S r
93 66 92 jca R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p = F r p F p / x S r
94 93 olcd R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p R F r F p = F r p F p / x S r
95 94 ex R We A R Se A x A S Po B p x A B q x A B r x A B F p = F q p F p / x S q F q = F r q F q / x S r F p R F r F p = F r p F p / x S r
96 53 58 63 95 ccased R We A R Se A x A S Po B p x A B q x A B r x A B F p R F q F p = F q p F p / x S q F q R F r F q = F r q F q / x S r F p R F r F p = F r p F p / x S r
97 44 46 96 syl2ani R We A R Se A x A S Po B p x A B q x A B r x A B p T q q T r F p R F r F p = F r p F p / x S r
98 1 2 weiunlem1 p T r p x A B r x A B F p R F r F p = F r p F p / x S r
99 98 biimpri p x A B r x A B F p R F r F p = F r p F p / x S r p T r
100 42 97 99 syl6an R We A R Se A x A S Po B p x A B q x A B r x A B p T q q T r p T r
101 40 100 jca R We A R Se A x A S Po B p x A B q x A B r x A B ¬ p T p p T q q T r p T r
102 101 ralrimivvva R We A R Se A x A S Po B p x A B q x A B r x A B ¬ p T p p T q q T r p T r
103 df-po T Po x A B p x A B q x A B r x A B ¬ p T p p T q q T r p T r
104 102 103 sylibr R We A R Se A x A S Po B T Po x A B