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Theorem nsgqusf1olem2 33169
Description: Lemma for nsgqusf1o 33171. (Contributed by Thierry Arnoux, 4-Aug-2024.)
Hypotheses
Ref Expression
nsgqusf1o.b 𝐡 = (Baseβ€˜πΊ)
nsgqusf1o.s 𝑆 = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}
nsgqusf1o.t 𝑇 = (SubGrpβ€˜π‘„)
nsgqusf1o.1 ≀ = (leβ€˜(toIncβ€˜π‘†))
nsgqusf1o.2 ≲ = (leβ€˜(toIncβ€˜π‘‡))
nsgqusf1o.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
nsgqusf1o.p βŠ• = (LSSumβ€˜πΊ)
nsgqusf1o.e 𝐸 = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))
nsgqusf1o.f 𝐹 = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
nsgqusf1o.n (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))
Assertion
Ref Expression
nsgqusf1olem2 (πœ‘ β†’ ran 𝐸 = 𝑇)
Distinct variable groups:   βŠ• ,π‘Ž,𝑓,β„Ž,π‘₯   𝐡,π‘Ž,𝑓,β„Ž,π‘₯   𝐸,π‘Ž,𝑓,β„Ž,π‘₯   𝑓,𝐹,β„Ž,π‘₯   𝐺,π‘Ž,𝑓,β„Ž,π‘₯   𝑁,π‘Ž,𝑓,β„Ž,π‘₯   𝑄,π‘Ž,𝑓,β„Ž,π‘₯   𝑆,π‘Ž,𝑓,β„Ž,π‘₯   𝑇,π‘Ž,𝑓,β„Ž,π‘₯   πœ‘,π‘Ž,𝑓,β„Ž,π‘₯
Allowed substitution hints:   𝐹(π‘Ž)   ≀ (π‘₯,𝑓,β„Ž,π‘Ž)   ≲ (π‘₯,𝑓,β„Ž,π‘Ž)

Proof of Theorem nsgqusf1olem2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 simpr 483 . . . . . . 7 (((πœ‘ ∧ β„Ž ∈ 𝑆) ∧ 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))) β†’ 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))
2 nsgqusf1o.s . . . . . . . . . 10 𝑆 = {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž}
32reqabi 3442 . . . . . . . . 9 (β„Ž ∈ 𝑆 ↔ (β„Ž ∈ (SubGrpβ€˜πΊ) ∧ 𝑁 βŠ† β„Ž))
4 nsgqusf1o.b . . . . . . . . . . . 12 𝐡 = (Baseβ€˜πΊ)
5 nsgqusf1o.t . . . . . . . . . . . 12 𝑇 = (SubGrpβ€˜π‘„)
6 nsgqusf1o.1 . . . . . . . . . . . 12 ≀ = (leβ€˜(toIncβ€˜π‘†))
7 nsgqusf1o.2 . . . . . . . . . . . 12 ≲ = (leβ€˜(toIncβ€˜π‘‡))
8 nsgqusf1o.q . . . . . . . . . . . 12 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
9 nsgqusf1o.p . . . . . . . . . . . 12 βŠ• = (LSSumβ€˜πΊ)
10 nsgqusf1o.e . . . . . . . . . . . 12 𝐸 = (β„Ž ∈ 𝑆 ↦ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))
11 nsgqusf1o.f . . . . . . . . . . . 12 𝐹 = (𝑓 ∈ 𝑇 ↦ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
12 nsgqusf1o.n . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))
134, 2, 5, 6, 7, 8, 9, 10, 11, 12nsgqusf1olem1 33168 . . . . . . . . . . 11 (((πœ‘ ∧ β„Ž ∈ (SubGrpβ€˜πΊ)) ∧ 𝑁 βŠ† β„Ž) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ 𝑇)
1413anasss 465 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (SubGrpβ€˜πΊ) ∧ 𝑁 βŠ† β„Ž)) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2835 . . . . . . . . 9 ((πœ‘ ∧ (β„Ž ∈ (SubGrpβ€˜πΊ) ∧ 𝑁 βŠ† β„Ž)) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ (SubGrpβ€˜π‘„))
163, 15sylan2b 592 . . . . . . . 8 ((πœ‘ ∧ β„Ž ∈ 𝑆) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ (SubGrpβ€˜π‘„))
1716adantr 479 . . . . . . 7 (((πœ‘ ∧ β„Ž ∈ 𝑆) ∧ 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ (SubGrpβ€˜π‘„))
181, 17eqeltrd 2825 . . . . . 6 (((πœ‘ ∧ β„Ž ∈ 𝑆) ∧ 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))) β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
1918r19.29an 3148 . . . . 5 ((πœ‘ ∧ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))) β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
20 sseq2 4000 . . . . . . . 8 (β„Ž = {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} β†’ (𝑁 βŠ† β„Ž ↔ 𝑁 βŠ† {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}))
2112adantr 479 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))
22 simpr 483 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
234, 8, 9, 21, 22nsgmgclem 33166 . . . . . . . 8 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∈ (SubGrpβ€˜πΊ))
245eleq2i 2817 . . . . . . . . 9 (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrpβ€˜π‘„))
25 nsgsubg 19112 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrpβ€˜πΊ) β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
2612, 25syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
274subgss 19081 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrpβ€˜πΊ) β†’ 𝑁 βŠ† 𝐡)
2826, 27syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑁 βŠ† 𝐡)
2928adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ 𝑁 βŠ† 𝐡)
3026ad2antrr 724 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
319grplsmid 33158 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrpβ€˜πΊ) ∧ π‘Ž ∈ 𝑁) β†’ ({π‘Ž} βŠ• 𝑁) = 𝑁)
3230, 31sylancom 586 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ ({π‘Ž} βŠ• 𝑁) = 𝑁)
3324biimpi 215 . . . . . . . . . . . . 13 (𝑓 ∈ 𝑇 β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
348nsgqus0 33165 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrpβ€˜πΊ) ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 ∈ 𝑓)
3512, 33, 34syl2an 594 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ 𝑁 ∈ 𝑓)
3635adantr 479 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ 𝑁 ∈ 𝑓)
3732, 36eqeltrd 2825 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓)
3829, 37ssrabdv 4064 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ 𝑁 βŠ† {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
3924, 38sylan2br 593 . . . . . . . 8 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 βŠ† {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3678 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∈ {β„Ž ∈ (SubGrpβ€˜πΊ) ∣ 𝑁 βŠ† β„Ž})
4140, 2eleqtrrdi 2836 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5237 . . . . . . . . 9 (β„Ž = {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} β†’ (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) = (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)))
4342rneqd 5935 . . . . . . . 8 (β„Ž = {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) = ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)))
4443eqeq2d 2736 . . . . . . 7 (β„Ž = {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} β†’ (𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ↔ 𝑓 = ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁))))
4544adantl 480 . . . . . 6 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ β„Ž = {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}) β†’ (𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ↔ 𝑓 = ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁))))
46 eqid 2725 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
4746subgss 19081 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrpβ€˜π‘„) β†’ 𝑓 βŠ† (Baseβ€˜π‘„))
4847adantl 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑓 βŠ† (Baseβ€˜π‘„))
4948sselda 3973 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝑖 ∈ (Baseβ€˜π‘„))
508a1i 11 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
514a1i 11 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝐡 = (Baseβ€˜πΊ))
52 ovexd 7448 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ (𝐺 ~QG 𝑁) ∈ V)
53 subgrcl 19085 . . . . . . . . . . . . . . 15 (𝑁 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
5426, 53syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐺 ∈ Grp)
5554ad2antrr 724 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝐺 ∈ Grp)
5650, 51, 52, 55qusbas 17521 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ (𝐡 / (𝐺 ~QG 𝑁)) = (Baseβ€˜π‘„))
5749, 56eleqtrrd 2828 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝑖 ∈ (𝐡 / (𝐺 ~QG 𝑁)))
58 elqsi 8782 . . . . . . . . . . 11 (𝑖 ∈ (𝐡 / (𝐺 ~QG 𝑁)) β†’ βˆƒπ‘₯ ∈ 𝐡 𝑖 = [π‘₯](𝐺 ~QG 𝑁))
5957, 58syl 17 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ βˆƒπ‘₯ ∈ 𝐡 𝑖 = [π‘₯](𝐺 ~QG 𝑁))
60 sneq 4635 . . . . . . . . . . . . . . . 16 (π‘Ž = π‘₯ β†’ {π‘Ž} = {π‘₯})
6160oveq1d 7428 . . . . . . . . . . . . . . 15 (π‘Ž = π‘₯ β†’ ({π‘Ž} βŠ• 𝑁) = ({π‘₯} βŠ• 𝑁))
6261eleq1d 2810 . . . . . . . . . . . . . 14 (π‘Ž = π‘₯ β†’ (({π‘Ž} βŠ• 𝑁) ∈ 𝑓 ↔ ({π‘₯} βŠ• 𝑁) ∈ 𝑓))
63 simplr 767 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ π‘₯ ∈ 𝐡)
64 simpr 483 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑖 = [π‘₯](𝐺 ~QG 𝑁))
6526ad4antr 730 . . . . . . . . . . . . . . . . 17 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
664, 9, 65, 63quslsm 33160 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ [π‘₯](𝐺 ~QG 𝑁) = ({π‘₯} βŠ• 𝑁))
6764, 66eqtrd 2765 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑖 = ({π‘₯} βŠ• 𝑁))
68 simpllr 774 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑖 ∈ 𝑓)
6967, 68eqeltrrd 2826 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3678 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
7170, 67jca 510 . . . . . . . . . . . 12 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)))
7271expl 456 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ ((π‘₯ ∈ 𝐡 ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∧ 𝑖 = ({π‘₯} βŠ• 𝑁))))
7372reximdv2 3154 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ (βˆƒπ‘₯ ∈ 𝐡 𝑖 = [π‘₯](𝐺 ~QG 𝑁) β†’ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁)))
7459, 73mpd 15 . . . . . . . . 9 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁))
75 simplr 767 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) β†’ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
7662elrab 3676 . . . . . . . . . . . 12 (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↔ (π‘₯ ∈ 𝐡 ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓))
7775, 76sylib 217 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) β†’ (π‘₯ ∈ 𝐡 ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓))
78 simpllr 774 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ π‘₯ ∈ 𝐡) ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓) β†’ 𝑖 = ({π‘₯} βŠ• 𝑁))
79 simpr 483 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ π‘₯ ∈ 𝐡) ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓) β†’ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)
8078, 79eqeltrd 2825 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ π‘₯ ∈ 𝐡) ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓) β†’ 𝑖 ∈ 𝑓)
8180anasss 465 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ (π‘₯ ∈ 𝐡 ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)) β†’ 𝑖 ∈ 𝑓)
8281adantllr 717 . . . . . . . . . . 11 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ (π‘₯ ∈ 𝐡 ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)) β†’ 𝑖 ∈ 𝑓)
8377, 82mpdan 685 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) β†’ 𝑖 ∈ 𝑓)
8483r19.29an 3148 . . . . . . . . 9 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁)) β†’ 𝑖 ∈ 𝑓)
8574, 84impbida 799 . . . . . . . 8 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ (𝑖 ∈ 𝑓 ↔ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁)))
86 eqid 2725 . . . . . . . . . 10 (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)) = (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁))
8786elrnmpt 5953 . . . . . . . . 9 (𝑖 ∈ V β†’ (𝑖 ∈ ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)) ↔ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁)))
8887elv 3469 . . . . . . . 8 (𝑖 ∈ ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)) ↔ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁))
8985, 88bitr4di 288 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ (𝑖 ∈ 𝑓 ↔ 𝑖 ∈ ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁))))
9089eqrdv 2723 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑓 = ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)))
9141, 45, 90rspcedvd 3605 . . . . 5 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))
9219, 91impbida 799 . . . 4 (πœ‘ β†’ (βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ↔ 𝑓 ∈ (SubGrpβ€˜π‘„)))
9392abbidv 2794 . . 3 (πœ‘ β†’ {𝑓 ∣ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrpβ€˜π‘„)})
9410rnmpt 5952 . . 3 ran 𝐸 = {𝑓 ∣ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))}
95 abid1 2862 . . 3 (SubGrpβ€˜π‘„) = {𝑓 ∣ 𝑓 ∈ (SubGrpβ€˜π‘„)}
9693, 94, 953eqtr4g 2790 . 2 (πœ‘ β†’ ran 𝐸 = (SubGrpβ€˜π‘„))
9796, 5eqtr4di 2783 1 (πœ‘ β†’ ran 𝐸 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆƒwrex 3060  {crab 3419  Vcvv 3463   βŠ† wss 3941  {csn 4625   ↦ cmpt 5227  ran crn 5674  β€˜cfv 6543  (class class class)co 7413  [cec 8716   / cqs 8717  Basecbs 17174  lecple 17234   /s cqus 17481  toInccipo 18513  Grpcgrp 18889  SubGrpcsubg 19074  NrmSGrpcnsg 19075   ~QG cqg 19076  LSSumclsm 19588
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 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
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 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-tpos 8225  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-ec 8720  df-qs 8724  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-2 12300  df-3 12301  df-4 12302  df-5 12303  df-6 12304  df-7 12305  df-8 12306  df-9 12307  df-n0 12498  df-z 12584  df-dec 12703  df-uz 12848  df-fz 13512  df-struct 17110  df-sets 17127  df-slot 17145  df-ndx 17157  df-base 17175  df-ress 17204  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-0g 17417  df-imas 17484  df-qus 17485  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-submnd 18735  df-grp 18892  df-minusg 18893  df-subg 19077  df-nsg 19078  df-eqg 19079  df-oppg 19296  df-lsm 19590
This theorem is referenced by:  nsgqusf1o  33171
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