Metamath Proof Explorer


Theorem mclsppslem

Description: The closure is closed under application of provable pre-statements. (Compare mclsax .) This theorem is what justifies the treatment of theorems as "equivalent" to axioms once they have been proven: the composition of one theorem in the proof of another yields a theorem. (Contributed by Mario Carneiro, 18-Jul-2016)

Ref Expression
Hypotheses mclspps.d D = mDV T
mclspps.e E = mEx T
mclspps.c C = mCls T
mclspps.1 φ T mFS
mclspps.2 φ K D
mclspps.3 φ B E
mclspps.j J = mPPSt T
mclspps.l L = mSubst T
mclspps.v V = mVR T
mclspps.h H = mVH T
mclspps.w W = mVars T
mclspps.4 φ M O P J
mclspps.5 φ S ran L
mclspps.6 φ x O S x K C B
mclspps.7 φ v V S H v K C B
mclspps.8 φ x M y a W S H x b W S H y a K b
mclsppslem.9 φ m o p mAx T
mclsppslem.10 φ s ran L
mclsppslem.11 φ s o ran H S -1 K C B
mclsppslem.12 φ z w z m w W s H z × W s H w M
Assertion mclsppslem φ s p S -1 K C B

Proof

Step Hyp Ref Expression
1 mclspps.d D = mDV T
2 mclspps.e E = mEx T
3 mclspps.c C = mCls T
4 mclspps.1 φ T mFS
5 mclspps.2 φ K D
6 mclspps.3 φ B E
7 mclspps.j J = mPPSt T
8 mclspps.l L = mSubst T
9 mclspps.v V = mVR T
10 mclspps.h H = mVH T
11 mclspps.w W = mVars T
12 mclspps.4 φ M O P J
13 mclspps.5 φ S ran L
14 mclspps.6 φ x O S x K C B
15 mclspps.7 φ v V S H v K C B
16 mclspps.8 φ x M y a W S H x b W S H y a K b
17 mclsppslem.9 φ m o p mAx T
18 mclsppslem.10 φ s ran L
19 mclsppslem.11 φ s o ran H S -1 K C B
20 mclsppslem.12 φ z w z m w W s H z × W s H w M
21 8 2 msubf s ran L s : E E
22 18 21 syl φ s : E E
23 eqid mAx T = mAx T
24 eqid mStat T = mStat T
25 23 24 maxsta T mFS mAx T mStat T
26 4 25 syl φ mAx T mStat T
27 eqid mPreSt T = mPreSt T
28 27 24 mstapst mStat T mPreSt T
29 26 28 sstrdi φ mAx T mPreSt T
30 29 17 sseldd φ m o p mPreSt T
31 1 2 27 elmpst m o p mPreSt T m D m -1 = m o E o Fin p E
32 30 31 sylib φ m D m -1 = m o E o Fin p E
33 32 simp3d φ p E
34 22 33 ffvelrnd φ s p E
35 fvco3 s : E E p E S s p = S s p
36 22 33 35 syl2anc φ S s p = S s p
37 8 msubco S ran L s ran L S s ran L
38 13 18 37 syl2anc φ S s ran L
39 8 2 msubf S ran L S : E E
40 13 39 syl φ S : E E
41 fco S : E E s : E E S s : E E
42 40 22 41 syl2anc φ S s : E E
43 42 ffnd φ S s Fn E
44 43 adantr φ c o S s Fn E
45 22 ffund φ Fun s
46 31 simp2bi m o p mPreSt T o E o Fin
47 30 46 syl φ o E o Fin
48 47 simpld φ o E
49 9 2 10 mvhf T mFS H : V E
50 frn H : V E ran H E
51 4 49 50 3syl φ ran H E
52 48 51 unssd φ o ran H E
53 22 fdmd φ dom s = E
54 52 53 sseqtrrd φ o ran H dom s
55 funimass3 Fun s o ran H dom s s o ran H S -1 K C B o ran H s -1 S -1 K C B
56 45 54 55 syl2anc φ s o ran H S -1 K C B o ran H s -1 S -1 K C B
57 19 56 mpbid φ o ran H s -1 S -1 K C B
58 cnvco S s -1 = s -1 S -1
59 58 imaeq1i S s -1 K C B = s -1 S -1 K C B
60 imaco s -1 S -1 K C B = s -1 S -1 K C B
61 59 60 eqtri S s -1 K C B = s -1 S -1 K C B
62 57 61 sseqtrrdi φ o ran H S s -1 K C B
63 62 unssad φ o S s -1 K C B
64 63 sselda φ c o c S s -1 K C B
65 elpreima S s Fn E c S s -1 K C B c E S s c K C B
66 65 simplbda S s Fn E c S s -1 K C B S s c K C B
67 44 64 66 syl2anc φ c o S s c K C B
68 43 adantr φ t V S s Fn E
69 62 unssbd φ ran H S s -1 K C B
70 69 adantr φ t V ran H S s -1 K C B
71 ffn H : V E H Fn V
72 4 49 71 3syl φ H Fn V
73 fnfvelrn H Fn V t V H t ran H
74 72 73 sylan φ t V H t ran H
75 70 74 sseldd φ t V H t S s -1 K C B
76 elpreima S s Fn E H t S s -1 K C B H t E S s H t K C B
77 76 simplbda S s Fn E H t S s -1 K C B S s H t K C B
78 68 75 77 syl2anc φ t V S s H t K C B
79 22 adantr φ c m d s : E E
80 4 49 syl φ H : V E
81 80 adantr φ c m d H : V E
82 32 simp1d φ m D m -1 = m
83 82 simpld φ m D
84 9 1 mdvval D = V × V I
85 difss V × V I V × V
86 84 85 eqsstri D V × V
87 83 86 sstrdi φ m V × V
88 87 ssbrd φ c m d c V × V d
89 88 imp φ c m d c V × V d
90 brxp c V × V d c V d V
91 89 90 sylib φ c m d c V d V
92 91 simpld φ c m d c V
93 81 92 ffvelrnd φ c m d H c E
94 fvco3 s : E E H c E S s H c = S s H c
95 79 93 94 syl2anc φ c m d S s H c = S s H c
96 95 fveq2d φ c m d W S s H c = W S s H c
97 4 adantr φ c m d T mFS
98 13 adantr φ c m d S ran L
99 79 93 ffvelrnd φ c m d s H c E
100 8 2 11 10 msubvrs T mFS S ran L s H c E W S s H c = u W s H c W S H u
101 97 98 99 100 syl3anc φ c m d W S s H c = u W s H c W S H u
102 96 101 eqtrd φ c m d W S s H c = u W s H c W S H u
103 102 eleq2d φ c m d a W S s H c a u W s H c W S H u
104 eliun a u W s H c W S H u u W s H c a W S H u
105 103 104 bitrdi φ c m d a W S s H c u W s H c a W S H u
106 91 simprd φ c m d d V
107 81 106 ffvelrnd φ c m d H d E
108 fvco3 s : E E H d E S s H d = S s H d
109 79 107 108 syl2anc φ c m d S s H d = S s H d
110 109 fveq2d φ c m d W S s H d = W S s H d
111 79 107 ffvelrnd φ c m d s H d E
112 8 2 11 10 msubvrs T mFS S ran L s H d E W S s H d = v W s H d W S H v
113 97 98 111 112 syl3anc φ c m d W S s H d = v W s H d W S H v
114 110 113 eqtrd φ c m d W S s H d = v W s H d W S H v
115 114 eleq2d φ c m d b W S s H d b v W s H d W S H v
116 eliun b v W s H d W S H v v W s H d b W S H v
117 115 116 bitrdi φ c m d b W S s H d v W s H d b W S H v
118 105 117 anbi12d φ c m d a W S s H c b W S s H d u W s H c a W S H u v W s H d b W S H v
119 reeanv u W s H c v W s H d a W S H u b W S H v u W s H c a W S H u v W s H d b W S H v
120 simpll φ c m d u W s H c v W s H d φ
121 brxp u W s H c × W s H d v u W s H c v W s H d
122 breq12 z = c w = d z m w c m d
123 simpl z = c w = d z = c
124 123 fveq2d z = c w = d H z = H c
125 124 fveq2d z = c w = d s H z = s H c
126 125 fveq2d z = c w = d W s H z = W s H c
127 simpr z = c w = d w = d
128 127 fveq2d z = c w = d H w = H d
129 128 fveq2d z = c w = d s H w = s H d
130 129 fveq2d z = c w = d W s H w = W s H d
131 126 130 xpeq12d z = c w = d W s H z × W s H w = W s H c × W s H d
132 131 sseq1d z = c w = d W s H z × W s H w M W s H c × W s H d M
133 122 132 imbi12d z = c w = d z m w W s H z × W s H w M c m d W s H c × W s H d M
134 133 spc2gv c V d V z w z m w W s H z × W s H w M c m d W s H c × W s H d M
135 134 el2v z w z m w W s H z × W s H w M c m d W s H c × W s H d M
136 20 135 syl φ c m d W s H c × W s H d M
137 136 imp φ c m d W s H c × W s H d M
138 137 ssbrd φ c m d u W s H c × W s H d v u M v
139 121 138 syl5bir φ c m d u W s H c v W s H d u M v
140 139 imp φ c m d u W s H c v W s H d u M v
141 vex u V
142 vex v V
143 breq12 x = u y = v x M y u M v
144 simpl x = u y = v x = u
145 144 fveq2d x = u y = v H x = H u
146 145 fveq2d x = u y = v S H x = S H u
147 146 fveq2d x = u y = v W S H x = W S H u
148 147 eleq2d x = u y = v a W S H x a W S H u
149 simpr x = u y = v y = v
150 149 fveq2d x = u y = v H y = H v
151 150 fveq2d x = u y = v S H y = S H v
152 151 fveq2d x = u y = v W S H y = W S H v
153 152 eleq2d x = u y = v b W S H y b W S H v
154 143 148 153 3anbi123d x = u y = v x M y a W S H x b W S H y u M v a W S H u b W S H v
155 154 anbi2d x = u y = v φ x M y a W S H x b W S H y φ u M v a W S H u b W S H v
156 155 imbi1d x = u y = v φ x M y a W S H x b W S H y a K b φ u M v a W S H u b W S H v a K b
157 141 142 156 16 vtocl2 φ u M v a W S H u b W S H v a K b
158 157 3exp2 φ u M v a W S H u b W S H v a K b
159 158 imp4b φ u M v a W S H u b W S H v a K b
160 120 140 159 syl2anc φ c m d u W s H c v W s H d a W S H u b W S H v a K b
161 160 rexlimdvva φ c m d u W s H c v W s H d a W S H u b W S H v a K b
162 119 161 syl5bir φ c m d u W s H c a W S H u v W s H d b W S H v a K b
163 118 162 sylbid φ c m d a W S s H c b W S s H d a K b
164 163 exp4b φ c m d a W S s H c b W S s H d a K b
165 164 3imp2 φ c m d a W S s H c b W S s H d a K b
166 1 2 3 4 5 6 23 8 9 10 11 17 38 67 78 165 mclsax φ S s p K C B
167 36 166 eqeltrrd φ S s p K C B
168 40 ffnd φ S Fn E
169 elpreima S Fn E s p S -1 K C B s p E S s p K C B
170 168 169 syl φ s p S -1 K C B s p E S s p K C B
171 34 167 170 mpbir2and φ s p S -1 K C B