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Theorem tgpconncompeqg 23479
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 8654 . . . . 5 (𝐴 ∈ 𝑋 β†’ [𝐴] ∼ = {𝑧 ∣ 𝐴 ∼ 𝑧})
21adantl 483 . . . 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 4038 . . . . . . . . . 10 {π‘₯ ∈ 𝒫 𝑋 ∣ ( 0 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} βŠ† 𝒫 𝑋
5 sspwuni 5061 . . . . . . . . . 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 3979 . . . . . . . 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 2733 . . . . . . . 8 (invgβ€˜πΊ) = (invgβ€˜πΊ)
11 eqid 2733 . . . . . . . 8 (+gβ€˜πΊ) = (+gβ€˜πΊ)
12 tgpconncompeqg.r . . . . . . . 8 ∼ = (𝐺 ~QG 𝑆)
139, 10, 11, 12eqgval 18984 . . . . . . 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 592 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∼ 𝑧 ↔ (𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝑧) ∈ 𝑆)))
15 simp2 1138 . . . . . 6 ((𝐴 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝑧) ∈ 𝑆) β†’ 𝑧 ∈ 𝑋)
1614, 15syl6bi 253 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∼ 𝑧 β†’ 𝑧 ∈ 𝑋))
1716abssdv 4026 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ {𝑧 ∣ 𝐴 ∼ 𝑧} βŠ† 𝑋)
182, 17eqsstrd 3983 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ βŠ† 𝑋)
19 simpr 486 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ 𝑋)
20 tgpgrp 23445 . . . . . . 7 (𝐺 ∈ TopGrp β†’ 𝐺 ∈ Grp)
21 tgpconncomp.z . . . . . . . 8 0 = (0gβ€˜πΊ)
229, 11, 21, 10grplinv 18805 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) = 0 )
2320, 22sylan 581 . . . . . 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 23449 . . . . . . . 8 (𝐺 ∈ TopGrp β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2625adantr 482 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2720adantr 482 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐺 ∈ Grp)
289, 21grpidcl 18783 . . . . . . . 8 (𝐺 ∈ Grp β†’ 0 ∈ 𝑋)
2927, 28syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ∈ 𝑋)
303conncompid 22798 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 0 ∈ 𝑋) β†’ 0 ∈ 𝑆)
3126, 29, 30syl2anc 585 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ∈ 𝑆)
3223, 31eqeltrd 2834 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (((invgβ€˜πΊ)β€˜π΄)(+gβ€˜πΊ)𝐴) ∈ 𝑆)
339, 10, 11, 12eqgval 18984 . . . . . 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 592 . . . . 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 1343 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∼ 𝐴)
36 elecg 8694 . . . . 5 ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
3719, 19, 36syl2anc 585 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 ∈ [𝐴] ∼ ↔ 𝐴 ∼ 𝐴))
3835, 37mpbird 257 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝐴 ∈ [𝐴] ∼ )
399, 12, 11eqglact 18986 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑆 βŠ† 𝑋 ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
407, 39mp3an2 1450 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
4120, 40sylan 581 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ = ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆))
4241oveq2d 7374 . . . 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 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
44 eqid 2733 . . . . . . 7 (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) = (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧))
4544, 9, 11, 24tgplacthmeo 23470 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) ∈ (𝐽Homeo𝐽))
46 hmeocn 23127 . . . . . 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 22279 . . . . . . 7 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 = βˆͺ 𝐽)
4926, 48syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑋 = βˆͺ 𝐽)
507, 49sseqtrid 3997 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ 𝑆 βŠ† βˆͺ 𝐽)
513conncompconn 22799 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 0 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
5226, 29, 51syl2anc 585 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
5343, 47, 50, 52connima 22792 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt ((𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑆)) ∈ Conn)
5442, 53eqeltrd 2834 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt [𝐴] ∼ ) ∈ Conn)
55 eqid 2733 . . . 4 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
5655conncompss 22800 . . 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 1372 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ [𝐴] ∼ βŠ† βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)})
58 elpwi 4568 . . . . . 6 (𝑦 ∈ 𝒫 𝑋 β†’ 𝑦 βŠ† 𝑋)
5944mptpreima 6191 . . . . . . . . . . 11 (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) = {𝑧 ∈ 𝑋 ∣ (𝐴(+gβ€˜πΊ)𝑧) ∈ 𝑦}
6059ssrab3 4041 . . . . . . . . . 10 (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) βŠ† 𝑋
6129adantr 482 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 0 ∈ 𝑋)
629, 11, 21grprid 18786 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
6320, 62sylan 581 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
6463adantr 482 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝐴(+gβ€˜πΊ) 0 ) = 𝐴)
65 simprrl 780 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝐴 ∈ 𝑦)
6664, 65eqeltrd 2834 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ (𝐴(+gβ€˜πΊ) 0 ) ∈ 𝑦)
67 oveq2 7366 . . . . . . . . . . . . 13 (𝑧 = 0 β†’ (𝐴(+gβ€˜πΊ)𝑧) = (𝐴(+gβ€˜πΊ) 0 ))
6867eleq1d 2819 . . . . . . . . . . . 12 (𝑧 = 0 β†’ ((𝐴(+gβ€˜πΊ)𝑧) ∈ 𝑦 ↔ (𝐴(+gβ€˜πΊ) 0 ) ∈ 𝑦))
6968, 59elrab2 3649 . . . . . . . . . . 11 ( 0 ∈ (β—‘(𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)) β€œ 𝑦) ↔ ( 0 ∈ 𝑋 ∧ (𝐴(+gβ€˜πΊ) 0 ) ∈ 𝑦))
7061, 66, 69sylanbrc 584 . . . . . . . . . 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 23128 . . . . . . . . . . . . 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 482 . . . . . . . . . . 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 770 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† 𝑋)
7549adantr 482 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑋 = βˆͺ 𝐽)
7674, 75sseqtrd 3985 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† βˆͺ 𝐽)
77 simprrr 781 . . . . . . . . . . 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 22792 . . . . . . . . . 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 22800 . . . . . . . . . 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 1467 . . . . . . . . 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 2733 . . . . . . . . . . . . . . . 16 (𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧))) = (𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))
8281, 9, 11, 10grplactcnv 18855 . . . . . . . . . . . . . . 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 581 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄):𝑋–1-1-onto→𝑋)
8581, 9grplactfval 18853 . . . . . . . . . . . . . . 15 (𝐴 ∈ 𝑋 β†’ ((𝑔 ∈ 𝑋 ↦ (𝑧 ∈ 𝑋 ↦ (𝑔(+gβ€˜πΊ)𝑧)))β€˜π΄) = (𝑧 ∈ 𝑋 ↦ (𝐴(+gβ€˜πΊ)𝑧)))
8685adantl 483 . . . . . . . . . . . . . 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 6780 . . . . . . . . . . . . 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 482 . . . . . . . . . . 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 6797 . . . . . . . . . . 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 6787 . . . . . . . . . . 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 6789 . . . . . . . . . . . 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 3986 . . . . . . . . . 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 7005 . . . . . . . . . 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 585 . . . . . . . . 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 482 . . . . . . . . 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 6159 . . . . . . . . 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 2791 . . . . . . . 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 3986 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 βŠ† 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))) β†’ 𝑦 βŠ† [𝐴] ∼ )
103102expr 458 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 βŠ† 𝑋) β†’ ((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
10458, 103sylan2 594 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) β†’ ((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
105104ralrimiva 3140 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴 ∈ 𝑋) β†’ βˆ€π‘¦ ∈ 𝒫 𝑋((𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn) β†’ 𝑦 βŠ† [𝐴] ∼ ))
106 eleq2w 2818 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ 𝑦))
107 oveq2 7366 . . . . . . 7 (π‘₯ = 𝑦 β†’ (𝐽 β†Ύt π‘₯) = (𝐽 β†Ύt 𝑦))
108107eleq1d 2819 . . . . . 6 (π‘₯ = 𝑦 β†’ ((𝐽 β†Ύt π‘₯) ∈ Conn ↔ (𝐽 β†Ύt 𝑦) ∈ Conn))
109106, 108anbi12d 632 . . . . 5 (π‘₯ = 𝑦 β†’ ((𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn)))
110109ralrab 3652 . . . 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 4901 . . 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 3962 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  {crab 3406   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866   class class class wbr 5106   ↦ cmpt 5189  β—‘ccnv 5633  dom cdm 5634   β€œ cima 5637  Fun wfun 6491  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  [cec 8649  Basecbs 17088  +gcplusg 17138   β†Ύt crest 17307  TopOpenctopn 17308  0gc0g 17326  Grpcgrp 18753  invgcminusg 18754   ~QG cqg 18929  TopOnctopon 22275   Cn ccn 22591  Conncconn 22778  Homeochmeo 23120  TopGrpctgp 23438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-ec 8653  df-map 8770  df-en 8887  df-fin 8890  df-fi 9352  df-rest 17309  df-0g 17328  df-topgen 17330  df-plusf 18501  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-eqg 18932  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-cn 22594  df-cnp 22595  df-conn 22779  df-tx 22929  df-hmeo 23122  df-tmd 23439  df-tgp 23440
This theorem is referenced by:  tgpconncomp  23480
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