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Theorem ghmqusker 19242
Description: A surjective group homomorphism 𝐹 from 𝐺 to 𝐻 induces an isomorphism 𝐽 from 𝑄 to 𝐻, where 𝑄 is the factor group of 𝐺 by 𝐹's kernel 𝐾. (Contributed by Thierry Arnoux, 15-Feb-2025.)
Hypotheses
Ref Expression
ghmqusker.1 0 = (0gβ€˜π»)
ghmqusker.f (πœ‘ β†’ 𝐹 ∈ (𝐺 GrpHom 𝐻))
ghmqusker.k 𝐾 = (◑𝐹 β€œ { 0 })
ghmqusker.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
ghmqusker.j 𝐽 = (π‘ž ∈ (Baseβ€˜π‘„) ↦ βˆͺ (𝐹 β€œ π‘ž))
ghmqusker.s (πœ‘ β†’ ran 𝐹 = (Baseβ€˜π»))
Assertion
Ref Expression
ghmqusker (πœ‘ β†’ 𝐽 ∈ (𝑄 GrpIso 𝐻))
Distinct variable groups:   𝐹,π‘ž   𝐺,π‘ž   𝐻,π‘ž   𝐽,π‘ž   𝐾,π‘ž   𝑄,π‘ž   πœ‘,π‘ž
Allowed substitution hint:   0 (π‘ž)

Proof of Theorem ghmqusker
Dummy variables π‘Ÿ π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghmqusker.1 . . 3 0 = (0gβ€˜π»)
2 ghmqusker.f . . 3 (πœ‘ β†’ 𝐹 ∈ (𝐺 GrpHom 𝐻))
3 ghmqusker.k . . 3 𝐾 = (◑𝐹 β€œ { 0 })
4 ghmqusker.q . . 3 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾))
5 ghmqusker.j . . 3 𝐽 = (π‘ž ∈ (Baseβ€˜π‘„) ↦ βˆͺ (𝐹 β€œ π‘ž))
61, 2, 3, 4, 5ghmquskerlem3 19241 . 2 (πœ‘ β†’ 𝐽 ∈ (𝑄 GrpHom 𝐻))
7 ghmgrp1 19176 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (𝐺 GrpHom 𝐻) β†’ 𝐺 ∈ Grp)
82, 7syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐺 ∈ Grp)
98ad4antr 730 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ 𝐺 ∈ Grp)
101ghmker 19200 . . . . . . . . . . . . . . . . . 18 (𝐹 ∈ (𝐺 GrpHom 𝐻) β†’ (◑𝐹 β€œ { 0 }) ∈ (NrmSGrpβ€˜πΊ))
112, 10syl 17 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (◑𝐹 β€œ { 0 }) ∈ (NrmSGrpβ€˜πΊ))
123, 11eqeltrid 2829 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝐾 ∈ (NrmSGrpβ€˜πΊ))
13 nsgsubg 19117 . . . . . . . . . . . . . . . 16 (𝐾 ∈ (NrmSGrpβ€˜πΊ) β†’ 𝐾 ∈ (SubGrpβ€˜πΊ))
1412, 13syl 17 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐾 ∈ (SubGrpβ€˜πΊ))
1514ad4antr 730 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ 𝐾 ∈ (SubGrpβ€˜πΊ))
16 eqid 2725 . . . . . . . . . . . . . . . . . . . . 21 (Baseβ€˜πΊ) = (Baseβ€˜πΊ)
17 eqid 2725 . . . . . . . . . . . . . . . . . . . . 21 (Baseβ€˜π») = (Baseβ€˜π»)
1816, 17ghmf 19178 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ (𝐺 GrpHom 𝐻) β†’ 𝐹:(Baseβ€˜πΊ)⟢(Baseβ€˜π»))
192, 18syl 17 . . . . . . . . . . . . . . . . . . 19 (πœ‘ β†’ 𝐹:(Baseβ€˜πΊ)⟢(Baseβ€˜π»))
2019ffnd 6718 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝐹 Fn (Baseβ€˜πΊ))
2120ad3antrrr 728 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ 𝐹 Fn (Baseβ€˜πΊ))
2221adantr 479 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ 𝐹 Fn (Baseβ€˜πΊ))
234a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ 𝑄 = (𝐺 /s (𝐺 ~QG 𝐾)))
24 eqidd 2726 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ (Baseβ€˜πΊ) = (Baseβ€˜πΊ))
25 ovexd 7451 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ (𝐺 ~QG 𝐾) ∈ V)
2623, 24, 25, 8qusbas 17526 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)) = (Baseβ€˜π‘„))
27 eqid 2725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐺 ~QG 𝐾) = (𝐺 ~QG 𝐾)
2816, 27eqger 19137 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐾 ∈ (SubGrpβ€˜πΊ) β†’ (𝐺 ~QG 𝐾) Er (Baseβ€˜πΊ))
2912, 13, 283syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 (πœ‘ β†’ (𝐺 ~QG 𝐾) Er (Baseβ€˜πΊ))
3029qsss 8795 . . . . . . . . . . . . . . . . . . . . . 22 (πœ‘ β†’ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)) βŠ† 𝒫 (Baseβ€˜πΊ))
3126, 30eqsstrrd 4012 . . . . . . . . . . . . . . . . . . . . 21 (πœ‘ β†’ (Baseβ€˜π‘„) βŠ† 𝒫 (Baseβ€˜πΊ))
3231sselda 3972 . . . . . . . . . . . . . . . . . . . 20 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ π‘Ÿ ∈ 𝒫 (Baseβ€˜πΊ))
3332elpwid 4607 . . . . . . . . . . . . . . . . . . 19 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ π‘Ÿ βŠ† (Baseβ€˜πΊ))
3433sselda 3972 . . . . . . . . . . . . . . . . . 18 (((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
3534adantr 479 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
3635adantr 479 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
37 simpr 483 . . . . . . . . . . . . . . . . . 18 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯))
3837eqeq1d 2727 . . . . . . . . . . . . . . . . 17 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ ((π½β€˜π‘Ÿ) = 0 ↔ (πΉβ€˜π‘₯) = 0 ))
3938biimpa 475 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ (πΉβ€˜π‘₯) = 0 )
40 fniniseg 7064 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Baseβ€˜πΊ) β†’ (π‘₯ ∈ (◑𝐹 β€œ { 0 }) ↔ (π‘₯ ∈ (Baseβ€˜πΊ) ∧ (πΉβ€˜π‘₯) = 0 )))
4140biimpar 476 . . . . . . . . . . . . . . . 16 ((𝐹 Fn (Baseβ€˜πΊ) ∧ (π‘₯ ∈ (Baseβ€˜πΊ) ∧ (πΉβ€˜π‘₯) = 0 )) β†’ π‘₯ ∈ (◑𝐹 β€œ { 0 }))
4222, 36, 39, 41syl12anc 835 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘₯ ∈ (◑𝐹 β€œ { 0 }))
4342, 3eleqtrrdi 2836 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘₯ ∈ 𝐾)
4427eqg0el 19142 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrpβ€˜πΊ)) β†’ ([π‘₯](𝐺 ~QG 𝐾) = 𝐾 ↔ π‘₯ ∈ 𝐾))
4544biimpar 476 . . . . . . . . . . . . . 14 (((𝐺 ∈ Grp ∧ 𝐾 ∈ (SubGrpβ€˜πΊ)) ∧ π‘₯ ∈ 𝐾) β†’ [π‘₯](𝐺 ~QG 𝐾) = 𝐾)
469, 15, 43, 45syl21anc 836 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ [π‘₯](𝐺 ~QG 𝐾) = 𝐾)
4729ad4antr 730 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ (𝐺 ~QG 𝐾) Er (Baseβ€˜πΊ))
48 simpr 483 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ π‘Ÿ ∈ (Baseβ€˜π‘„))
4926adantr 479 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)) = (Baseβ€˜π‘„))
5048, 49eleqtrrd 2828 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ π‘Ÿ ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)))
5150ad3antrrr 728 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘Ÿ ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)))
52 simpllr 774 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘₯ ∈ π‘Ÿ)
53 qsel 8813 . . . . . . . . . . . . . 14 (((𝐺 ~QG 𝐾) Er (Baseβ€˜πΊ) ∧ π‘Ÿ ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)) ∧ π‘₯ ∈ π‘Ÿ) β†’ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾))
5447, 51, 52, 53syl3anc 1368 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾))
55 eqid 2725 . . . . . . . . . . . . . . . 16 (0gβ€˜πΊ) = (0gβ€˜πΊ)
5616, 27, 55eqgid 19139 . . . . . . . . . . . . . . 15 (𝐾 ∈ (SubGrpβ€˜πΊ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = 𝐾)
5714, 56syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = 𝐾)
5857ad4antr 730 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = 𝐾)
5946, 54, 583eqtr4d 2775 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘Ÿ = [(0gβ€˜πΊ)](𝐺 ~QG 𝐾))
604, 55qus0 19148 . . . . . . . . . . . . . . 15 (𝐾 ∈ (NrmSGrpβ€˜πΊ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = (0gβ€˜π‘„))
6112, 60syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = (0gβ€˜π‘„))
6261ad3antrrr 728 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = (0gβ€˜π‘„))
6362adantr 479 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) = (0gβ€˜π‘„))
6459, 63eqtrd 2765 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ (π½β€˜π‘Ÿ) = 0 ) β†’ π‘Ÿ = (0gβ€˜π‘„))
6562eqeq2d 2736 . . . . . . . . . . . . . 14 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ (π‘Ÿ = [(0gβ€˜πΊ)](𝐺 ~QG 𝐾) ↔ π‘Ÿ = (0gβ€˜π‘„)))
6665biimpar 476 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ π‘Ÿ = [(0gβ€˜πΊ)](𝐺 ~QG 𝐾))
6766fveq2d 6896 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ (π½β€˜π‘Ÿ) = (π½β€˜[(0gβ€˜πΊ)](𝐺 ~QG 𝐾)))
682adantr 479 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ 𝐹 ∈ (𝐺 GrpHom 𝐻))
6968ad3antrrr 728 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ 𝐹 ∈ (𝐺 GrpHom 𝐻))
7016, 55grpidcl 18926 . . . . . . . . . . . . . . 15 (𝐺 ∈ Grp β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
718, 70syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
7271ad4antr 730 . . . . . . . . . . . . 13 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ (0gβ€˜πΊ) ∈ (Baseβ€˜πΊ))
731, 69, 3, 4, 5, 72ghmquskerlem1 19238 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ (π½β€˜[(0gβ€˜πΊ)](𝐺 ~QG 𝐾)) = (πΉβ€˜(0gβ€˜πΊ)))
7455, 1ghmid 19180 . . . . . . . . . . . . . 14 (𝐹 ∈ (𝐺 GrpHom 𝐻) β†’ (πΉβ€˜(0gβ€˜πΊ)) = 0 )
752, 74syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ (πΉβ€˜(0gβ€˜πΊ)) = 0 )
7675ad4antr 730 . . . . . . . . . . . 12 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ (πΉβ€˜(0gβ€˜πΊ)) = 0 )
7767, 73, 763eqtrd 2769 . . . . . . . . . . 11 (((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ (π½β€˜π‘Ÿ) = 0 )
7864, 77impbida 799 . . . . . . . . . 10 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ ((π½β€˜π‘Ÿ) = 0 ↔ π‘Ÿ = (0gβ€˜π‘„)))
791, 68, 3, 4, 5, 48ghmquskerlem2 19240 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ βˆƒπ‘₯ ∈ π‘Ÿ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯))
8078, 79r19.29a 3152 . . . . . . . . 9 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ ((π½β€˜π‘Ÿ) = 0 ↔ π‘Ÿ = (0gβ€˜π‘„)))
8180pm5.32da 577 . . . . . . . 8 (πœ‘ β†’ ((π‘Ÿ ∈ (Baseβ€˜π‘„) ∧ (π½β€˜π‘Ÿ) = 0 ) ↔ (π‘Ÿ ∈ (Baseβ€˜π‘„) ∧ π‘Ÿ = (0gβ€˜π‘„))))
82 simpr 483 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ π‘Ÿ = (0gβ€˜π‘„))
834qusgrp 19145 . . . . . . . . . . . . . 14 (𝐾 ∈ (NrmSGrpβ€˜πΊ) β†’ 𝑄 ∈ Grp)
8412, 83syl 17 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑄 ∈ Grp)
85 eqid 2725 . . . . . . . . . . . . . 14 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
86 eqid 2725 . . . . . . . . . . . . . 14 (0gβ€˜π‘„) = (0gβ€˜π‘„)
8785, 86grpidcl 18926 . . . . . . . . . . . . 13 (𝑄 ∈ Grp β†’ (0gβ€˜π‘„) ∈ (Baseβ€˜π‘„))
8884, 87syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (0gβ€˜π‘„) ∈ (Baseβ€˜π‘„))
8988adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ (0gβ€˜π‘„) ∈ (Baseβ€˜π‘„))
9082, 89eqeltrd 2825 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ = (0gβ€˜π‘„)) β†’ π‘Ÿ ∈ (Baseβ€˜π‘„))
9190ex 411 . . . . . . . . 9 (πœ‘ β†’ (π‘Ÿ = (0gβ€˜π‘„) β†’ π‘Ÿ ∈ (Baseβ€˜π‘„)))
9291pm4.71rd 561 . . . . . . . 8 (πœ‘ β†’ (π‘Ÿ = (0gβ€˜π‘„) ↔ (π‘Ÿ ∈ (Baseβ€˜π‘„) ∧ π‘Ÿ = (0gβ€˜π‘„))))
9381, 92bitr4d 281 . . . . . . 7 (πœ‘ β†’ ((π‘Ÿ ∈ (Baseβ€˜π‘„) ∧ (π½β€˜π‘Ÿ) = 0 ) ↔ π‘Ÿ = (0gβ€˜π‘„)))
942adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘ž ∈ (Baseβ€˜π‘„)) β†’ 𝐹 ∈ (𝐺 GrpHom 𝐻))
9594imaexd 7922 . . . . . . . . . . 11 ((πœ‘ ∧ π‘ž ∈ (Baseβ€˜π‘„)) β†’ (𝐹 β€œ π‘ž) ∈ V)
9695uniexd 7745 . . . . . . . . . 10 ((πœ‘ ∧ π‘ž ∈ (Baseβ€˜π‘„)) β†’ βˆͺ (𝐹 β€œ π‘ž) ∈ V)
975a1i 11 . . . . . . . . . 10 (πœ‘ β†’ 𝐽 = (π‘ž ∈ (Baseβ€˜π‘„) ↦ βˆͺ (𝐹 β€œ π‘ž)))
9821, 35fnfvelrnd 7087 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ (πΉβ€˜π‘₯) ∈ ran 𝐹)
99 ghmqusker.s . . . . . . . . . . . . . 14 (πœ‘ β†’ ran 𝐹 = (Baseβ€˜π»))
10099ad3antrrr 728 . . . . . . . . . . . . 13 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ ran 𝐹 = (Baseβ€˜π»))
10198, 100eleqtrd 2827 . . . . . . . . . . . 12 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ (πΉβ€˜π‘₯) ∈ (Baseβ€˜π»))
10237, 101eqeltrd 2825 . . . . . . . . . . 11 ((((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) ∧ π‘₯ ∈ π‘Ÿ) ∧ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯)) β†’ (π½β€˜π‘Ÿ) ∈ (Baseβ€˜π»))
103102, 79r19.29a 3152 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ (π½β€˜π‘Ÿ) ∈ (Baseβ€˜π»))
10496, 97, 103fmpt2d 7129 . . . . . . . . 9 (πœ‘ β†’ 𝐽:(Baseβ€˜π‘„)⟢(Baseβ€˜π»))
105104ffnd 6718 . . . . . . . 8 (πœ‘ β†’ 𝐽 Fn (Baseβ€˜π‘„))
106 fniniseg 7064 . . . . . . . 8 (𝐽 Fn (Baseβ€˜π‘„) β†’ (π‘Ÿ ∈ (◑𝐽 β€œ { 0 }) ↔ (π‘Ÿ ∈ (Baseβ€˜π‘„) ∧ (π½β€˜π‘Ÿ) = 0 )))
107105, 106syl 17 . . . . . . 7 (πœ‘ β†’ (π‘Ÿ ∈ (◑𝐽 β€œ { 0 }) ↔ (π‘Ÿ ∈ (Baseβ€˜π‘„) ∧ (π½β€˜π‘Ÿ) = 0 )))
108 velsn 4640 . . . . . . . 8 (π‘Ÿ ∈ {(0gβ€˜π‘„)} ↔ π‘Ÿ = (0gβ€˜π‘„))
109108a1i 11 . . . . . . 7 (πœ‘ β†’ (π‘Ÿ ∈ {(0gβ€˜π‘„)} ↔ π‘Ÿ = (0gβ€˜π‘„)))
11093, 107, 1093bitr4d 310 . . . . . 6 (πœ‘ β†’ (π‘Ÿ ∈ (◑𝐽 β€œ { 0 }) ↔ π‘Ÿ ∈ {(0gβ€˜π‘„)}))
111110eqrdv 2723 . . . . 5 (πœ‘ β†’ (◑𝐽 β€œ { 0 }) = {(0gβ€˜π‘„)})
11285, 17, 86, 1kerf1ghm 19205 . . . . . 6 (𝐽 ∈ (𝑄 GrpHom 𝐻) β†’ (𝐽:(Baseβ€˜π‘„)–1-1β†’(Baseβ€˜π») ↔ (◑𝐽 β€œ { 0 }) = {(0gβ€˜π‘„)}))
113112biimpar 476 . . . . 5 ((𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ (◑𝐽 β€œ { 0 }) = {(0gβ€˜π‘„)}) β†’ 𝐽:(Baseβ€˜π‘„)–1-1β†’(Baseβ€˜π»))
1146, 111, 113syl2anc 582 . . . 4 (πœ‘ β†’ 𝐽:(Baseβ€˜π‘„)–1-1β†’(Baseβ€˜π»))
115 f1f1orn 6845 . . . 4 (𝐽:(Baseβ€˜π‘„)–1-1β†’(Baseβ€˜π») β†’ 𝐽:(Baseβ€˜π‘„)–1-1-ontoβ†’ran 𝐽)
116114, 115syl 17 . . 3 (πœ‘ β†’ 𝐽:(Baseβ€˜π‘„)–1-1-ontoβ†’ran 𝐽)
117 simpr 483 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ π‘₯ ∈ (Baseβ€˜πΊ))
118 ovex 7449 . . . . . . . . . . 11 (𝐺 ~QG 𝐾) ∈ V
119118ecelqsi 8790 . . . . . . . . . 10 (π‘₯ ∈ (Baseβ€˜πΊ) β†’ [π‘₯](𝐺 ~QG 𝐾) ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)))
120117, 119syl 17 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ [π‘₯](𝐺 ~QG 𝐾) ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)))
12126adantr 479 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)) = (Baseβ€˜π‘„))
122120, 121eleqtrd 2827 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ [π‘₯](𝐺 ~QG 𝐾) ∈ (Baseβ€˜π‘„))
123 elqsi 8787 . . . . . . . . 9 (π‘Ÿ ∈ ((Baseβ€˜πΊ) / (𝐺 ~QG 𝐾)) β†’ βˆƒπ‘₯ ∈ (Baseβ€˜πΊ)π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾))
12450, 123syl 17 . . . . . . . 8 ((πœ‘ ∧ π‘Ÿ ∈ (Baseβ€˜π‘„)) β†’ βˆƒπ‘₯ ∈ (Baseβ€˜πΊ)π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾))
125 simpr 483 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) ∧ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾)) β†’ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾))
126125fveq2d 6896 . . . . . . . . . . 11 (((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) ∧ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾)) β†’ (π½β€˜π‘Ÿ) = (π½β€˜[π‘₯](𝐺 ~QG 𝐾)))
1272adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ 𝐹 ∈ (𝐺 GrpHom 𝐻))
1281, 127, 3, 4, 5, 117ghmquskerlem1 19238 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) β†’ (π½β€˜[π‘₯](𝐺 ~QG 𝐾)) = (πΉβ€˜π‘₯))
129128adantr 479 . . . . . . . . . . 11 (((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) ∧ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾)) β†’ (π½β€˜[π‘₯](𝐺 ~QG 𝐾)) = (πΉβ€˜π‘₯))
130126, 129eqtrd 2765 . . . . . . . . . 10 (((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ)) ∧ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾)) β†’ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯))
1311303impa 1107 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾)) β†’ (π½β€˜π‘Ÿ) = (πΉβ€˜π‘₯))
132131eqeq1d 2727 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ (Baseβ€˜πΊ) ∧ π‘Ÿ = [π‘₯](𝐺 ~QG 𝐾)) β†’ ((π½β€˜π‘Ÿ) = 𝑦 ↔ (πΉβ€˜π‘₯) = 𝑦))
133122, 124, 132rexxfrd2 5407 . . . . . . 7 (πœ‘ β†’ (βˆƒπ‘Ÿ ∈ (Baseβ€˜π‘„)(π½β€˜π‘Ÿ) = 𝑦 ↔ βˆƒπ‘₯ ∈ (Baseβ€˜πΊ)(πΉβ€˜π‘₯) = 𝑦))
134 fvelrnb 6954 . . . . . . . 8 (𝐽 Fn (Baseβ€˜π‘„) β†’ (𝑦 ∈ ran 𝐽 ↔ βˆƒπ‘Ÿ ∈ (Baseβ€˜π‘„)(π½β€˜π‘Ÿ) = 𝑦))
135105, 134syl 17 . . . . . . 7 (πœ‘ β†’ (𝑦 ∈ ran 𝐽 ↔ βˆƒπ‘Ÿ ∈ (Baseβ€˜π‘„)(π½β€˜π‘Ÿ) = 𝑦))
136 fvelrnb 6954 . . . . . . . 8 (𝐹 Fn (Baseβ€˜πΊ) β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯ ∈ (Baseβ€˜πΊ)(πΉβ€˜π‘₯) = 𝑦))
13720, 136syl 17 . . . . . . 7 (πœ‘ β†’ (𝑦 ∈ ran 𝐹 ↔ βˆƒπ‘₯ ∈ (Baseβ€˜πΊ)(πΉβ€˜π‘₯) = 𝑦))
138133, 135, 1373bitr4rd 311 . . . . . 6 (πœ‘ β†’ (𝑦 ∈ ran 𝐹 ↔ 𝑦 ∈ ran 𝐽))
139138eqrdv 2723 . . . . 5 (πœ‘ β†’ ran 𝐹 = ran 𝐽)
140139, 99eqtr3d 2767 . . . 4 (πœ‘ β†’ ran 𝐽 = (Baseβ€˜π»))
141140f1oeq3d 6831 . . 3 (πœ‘ β†’ (𝐽:(Baseβ€˜π‘„)–1-1-ontoβ†’ran 𝐽 ↔ 𝐽:(Baseβ€˜π‘„)–1-1-ontoβ†’(Baseβ€˜π»)))
142116, 141mpbid 231 . 2 (πœ‘ β†’ 𝐽:(Baseβ€˜π‘„)–1-1-ontoβ†’(Baseβ€˜π»))
14385, 17isgim 19220 . 2 (𝐽 ∈ (𝑄 GrpIso 𝐻) ↔ (𝐽 ∈ (𝑄 GrpHom 𝐻) ∧ 𝐽:(Baseβ€˜π‘„)–1-1-ontoβ†’(Baseβ€˜π»)))
1446, 142, 143sylanbrc 581 1 (πœ‘ β†’ 𝐽 ∈ (𝑄 GrpIso 𝐻))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060  Vcvv 3463  π’« cpw 4598  {csn 4624  βˆͺ cuni 4903   ↦ cmpt 5226  β—‘ccnv 5671  ran crn 5673   β€œ cima 5675   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7416   Er wer 8720  [cec 8721   / cqs 8722  Basecbs 17179  0gc0g 17420   /s cqus 17486  Grpcgrp 18894  SubGrpcsubg 19079  NrmSGrpcnsg 19080   ~QG cqg 19081   GrpHom cghm 19171   GrpIso cgim 19215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-ec 8725  df-qs 8729  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-sup 9465  df-inf 9466  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-sca 17248  df-vsca 17249  df-ip 17250  df-tset 17251  df-ple 17252  df-ds 17254  df-0g 17422  df-imas 17489  df-qus 17490  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18740  df-grp 18897  df-minusg 18898  df-sbg 18899  df-subg 19082  df-nsg 19083  df-eqg 19084  df-ghm 19172  df-gim 19217
This theorem is referenced by:  gicqusker  19243  lmhmqusker  33176  rhmqusker  33190  aks6d1c6lem5  41705
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