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Theorem tgpconncompeqg 23837
Description: The connected component containing 𝐴 is the left coset of the identity component containing 𝐴. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypotheses
Ref Expression
tgpconncomp.x 𝑋 = (Baseβ€˜πΊ)
tgpconncomp.z 0 = (0gβ€˜πΊ)
tgpconncomp.j 𝐽 = (TopOpenβ€˜πΊ)
tgpconncomp.s 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
tgpconncompeqg.r ∼ = (𝐺 ~QG 𝑆)
Assertion
Ref Expression
tgpconncompeqg ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)})
Distinct variable groups:   π‘₯, 0   π‘₯,𝐴   π‘₯,𝐽   π‘₯,𝐺   π‘₯,𝑋
Allowed substitution hints:   ∼ (π‘₯)   𝑆(π‘₯)
Proof of Theorem tgpconncompeqg
Dummy variables 𝑦 𝑧 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfec2 8709 . . . . 5 (𝐴 ∈ 𝑋 β†’ [𝐴] ∼ = {𝑧 ∣ 𝐴 ∼ 𝑧})
21adantl 481 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = {𝑧 ∣ 𝐴 ∼ 𝑧})
3 tgpconncomp.s . . . . . . . . 9 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
4 ssrab2 4078 . . . . . . . . . 10 {π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋
5 sspwuni 5104 . . . . . . . . . 10 ({π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋 ↔ βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝑋)
64, 5mpbi 229 . . . . . . . . 9 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝑋
73, 6eqsstri 4017 . . . . . . . 8 𝑆 βŠ† 𝑋
87a1i 11 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† 𝑋)
9 tgpconncomp.x . . . . . . . 8 𝑋 = (Baseβ€˜πΊ)
10 eqid 2731 . . . . . . . 8 (invgβ€˜πΊ) = (invgβ€˜πΊ)
11 eqid 2731 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
12 tgpconncompeqg.r . . . . . . . 8 ∼ = (𝐺 ~QG 𝑆)
139, 10, 11, 12eqgval 19094 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 βŠ† 𝑋) β†’ (𝐴 ∼ 𝑧 ↔ (𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝑧) ∈ 𝑆)))
148, 13syldan 590 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∼ 𝑧 ↔ (𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝑧) ∈ 𝑆)))
15 simp2 1136 . . . . . 6 ((𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝑧) ∈ 𝑆) β†’ 𝑧 ∈ 𝑋)
1614, 15syl6bi 252 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∼ 𝑧 β†’ 𝑧 ∈ 𝑋))
1716abssdv 4066 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ {𝑧 ∣ 𝐴 ∼ 𝑧} βŠ† 𝑋)
182, 17eqsstrd 4021 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ βŠ† 𝑋)
19 simpr 484 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
20 tgpgrp 23803 . . . . . . 7 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
21 tgpconncomp.z . . . . . . . 8 0 = (0gβ€˜πΊ)
229, 11, 21, 10grplinv 18911 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) = 0 )
2320, 22sylan 579 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) = 0 )
24 tgpconncomp.j . . . . . . . . 9 𝐽 = (TopOpenβ€˜πΊ)
2524, 9tgptopon 23807 . . . . . . . 8 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2625adantr 480 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2720adantr 480 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐺 ∈ Grp)
289, 21grpidcl 18887 . . . . . . . 8 (𝐺 ∈ Grp β†’ 0 ∈ 𝑋)
2927, 28syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ∈ 𝑋)
303conncompid 23156 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 0 ∈ 𝑋) β†’ 0 ∈ 𝑆)
3126, 29, 30syl2anc 583 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ∈ 𝑆)
3223, 31eqeltrd 2832 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) ∈ 𝑆)
339, 10, 11, 12eqgval 19094 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 βŠ† 𝑋) β†’ (𝐴 ∼ 𝐴 ↔ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) ∈ 𝑆)))
348, 33syldan 590 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∼ 𝐴 ↔ (𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) ∈ 𝑆)))
3519, 19, 32, 34mpbir3and 1341 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∼ 𝐴)
36 elecg 8749 . . . . 5 ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
3719, 19, 36syl2anc 583 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
3835, 37mpbird 256 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ [𝐴] ∼ )
399, 12, 11eqglact 19096 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
407, 39mp3an2 1448 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
4120, 40sylan 579 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
4241oveq2d 7428 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt [𝐴] ∼ ) = (𝐽 β†Ύt ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆)))
43 eqid 2731 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
44 eqid 2731 . . . . . . 7 (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧))
4544, 9, 11, 24tgplacthmeo 23828 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽))
46 hmeocn 23485 . . . . . 6 ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽) β†’ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽 Cn 𝐽))
4745, 46syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽 Cn 𝐽))
48 toponuni 22637 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
4926, 48syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
507, 49sseqtrid 4035 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
513conncompconn 23157 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 0 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
5226, 29, 51syl2anc 583 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
5343, 47, 50, 52connima 23150 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆)) ∈ Conn)
5442, 53eqeltrd 2832 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt [𝐴] ∼ ) ∈ Conn)
55 eqid 2731 . . . 4 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
5655conncompss 23158 . . 3 (([𝐴] ∼ βŠ† 𝑋 ∧ 𝐴 ∈ [𝐴] ∼ ∧ (𝐽 β†Ύt [𝐴] ∼ ) ∈ Conn) β†’ [𝐴] ∼ βŠ† βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)})
5718, 38, 54, 56syl3anc 1370 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ βŠ† βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)})
58 elpwi 4610 . . . . . 6 (𝑦 ∈ 𝒫 𝑋 β†’ 𝑦 βŠ† 𝑋)
5944mptpreima 6238 . . . . . . . . . . 11 (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) = {𝑧 ∈ 𝑋 ∣ (𝐴(+gβ€˜πΊ)𝑧) ∈ 𝑦}
6059ssrab3 4081 . . . . . . . . . 10 (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑋
6129adantr 480 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 0 ∈ 𝑋)
629, 11, 21grprid 18890 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
6320, 62sylan 579 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
6463adantr 480 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
65 simprrl 778 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝐴 ∈ 𝑦)
6664, 65eqeltrd 2832 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝐴(+gβ€˜πΊ) 0 ) ∈ 𝑦)
67 oveq2 7420 . . . . . . . . . . . . 13 (𝑧 = 0 β†’ (𝐴(+gβ€˜πΊ)𝑧) = (𝐴(+gβ€˜πΊ) 0 ))
6867eleq1d 2817 . . . . . . . . . . . 12 (𝑧 = 0 β†’ ((𝐴(+gβ€˜πΊ)𝑧) ∈ 𝑦 ↔ (𝐴(+gβ€˜πΊ) 0 ) ∈ 𝑦))
6968, 59elrab2 3687 . . . . . . . . . . 11 ( 0 ∈ (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) ↔ ( 0 ∈ 𝑋 ∧ (𝐴(+gβ€˜πΊ) 0 ) ∈ 𝑦))
7061, 66, 69sylanbrc 582 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 0 ∈ (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦))
71 hmeocnvcn 23486 . . . . . . . . . . . . 13 ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽) β†’ β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽 Cn 𝐽))
7245, 71syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽 Cn 𝐽))
7372adantr 480 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽 Cn 𝐽))
74 simprl 768 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† 𝑋)
7549adantr 480 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑋 = βˆͺ 𝐽)
7674, 75sseqtrd 4023 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† βˆͺ 𝐽)
77 simprrr 779 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝐽 β†Ύt 𝑦) ∈ Conn)
7843, 73, 76, 77connima 23150 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝐽 β†Ύt (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦)) ∈ Conn)
793conncompss 23158 . . . . . . . . . 10 (((β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑋 ∧ 0 ∈ (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) ∧ (𝐽 β†Ύt (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦)) ∈ Conn) β†’ (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑆)
8060, 70, 78, 79mp3an2i 1465 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑆)
81 eqid 2731 . . . . . . . . . . . . . . . 16 (𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧))) = (𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))
8281, 9, 11, 10grplactcnv 18963 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄):𝑋–1-1-onto→𝑋 ∧ β—‘((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄) = ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜((invgβ€˜πΊ)β€˜π΄))))
8320, 82sylan 579 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄):𝑋–1-1-onto→𝑋 ∧ β—‘((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄) = ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜((invgβ€˜πΊ)β€˜π΄))))
8483simpld 494 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄):𝑋–1-1-onto→𝑋)
8581, 9grplactfval 18961 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝑋 β†’ ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄) = (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)))
8685adantl 481 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄) = (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)))
8786f1oeq1d 6829 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄):𝑋–1-1-onto→𝑋 ↔ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋))
8884, 87mpbid 231 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋)
8988adantr 480 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋)
90 f1ocnv 6846 . . . . . . . . . . 11 ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋 β†’ β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋)
91 f1ofun 6836 . . . . . . . . . . 11 (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋 β†’ Fun β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)))
9289, 90, 913syl 18 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ Fun β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)))
93 f1odm 6838 . . . . . . . . . . . 12 (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)):𝑋–1-1-onto→𝑋 β†’ dom β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) = 𝑋)
9489, 90, 933syl 18 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ dom β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) = 𝑋)
9574, 94sseqtrrd 4024 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† dom β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)))
96 funimass3 7056 . . . . . . . . . 10 ((Fun β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∧ 𝑦 βŠ† dom β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧))) β†’ ((β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑆 ↔ 𝑦 βŠ† (β—‘β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆)))
9792, 95, 96syl2anc 583 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ ((β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑆 ↔ 𝑦 βŠ† (β—‘β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆)))
9880, 97mpbid 231 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† (β—‘β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
9941adantr 480 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
100 imacnvcnv 6206 . . . . . . . . 9 (β—‘β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆) = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆)
10199, 100eqtr4di 2789 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ [𝐴] ∼ = (β—‘β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
10298, 101sseqtrrd 4024 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† [𝐴] ∼ )
103102expr 456 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 βŠ† 𝑋) β†’ ((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
10458, 103sylan2 592 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) β†’ ((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
105104ralrimiva 3145 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ βˆ€π‘¦ ∈ 𝒫 𝑋((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
106 eleq2w 2816 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ 𝑦))
107 oveq2 7420 . . . . . . 7 (π‘₯ = 𝑦 β†’ (𝐽 β†Ύt π‘₯) = (𝐽 β†Ύt 𝑦))
108107eleq1d 2817 . . . . . 6 (π‘₯ = 𝑦 β†’ ((𝐽 β†Ύt π‘₯) ∈ Conn ↔ (𝐽 β†Ύt 𝑦) ∈ Conn))
109106, 108anbi12d 630 . . . . 5 (π‘₯ = 𝑦 β†’ ((𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn)))
110109ralrab 3690 . . . 4 (βˆ€π‘¦ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦 βŠ† [𝐴] ∼ ↔ βˆ€π‘¦ ∈ 𝒫 𝑋((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
111105, 110sylibr 233 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ βˆ€π‘¦ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦 βŠ† [𝐴] ∼ )
112 unissb 4944 . . 3 (βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† [𝐴] ∼ ↔ βˆ€π‘¦ ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦 βŠ† [𝐴] ∼ )
113111, 112sylibr 233 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† [𝐴] ∼ )
11457, 113eqssd 4000 1 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  {cab 2708  βˆ€wral 3060  {crab 3431   βŠ† wss 3949  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677   β€œ cima 5680  Fun wfun 6538  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7412  [cec 8704  Basecbs 17149  +gcplusg 17202   β†Ύt crest 17371  TopOpenctopn 17372  0gc0g 17390  Grpcgrp 18856  invgcminusg 18857   ~QG cqg 19039  TopOnctopon 22633   Cn ccn 22949  Conncconn 23136  Homeochmeo 23478  TopGrpctgp 23796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-ec 8708  df-map 8825  df-en 8943  df-fin 8946  df-fi 9409  df-rest 17373  df-0g 17392  df-topgen 17394  df-plusf 18565  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-eqg 19042  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-cld 22744  df-cn 22952  df-cnp 22953  df-conn 23137  df-tx 23287  df-hmeo 23480  df-tmd 23797  df-tgp 23798
This theorem is referenced by:  tgpconncomp  23838
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