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Theorem nsgqusf1olem2 33064
Description: Lemma for nsgqusf1o 33066. (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 484 . . . . . . 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 3449 . . . . . . . . 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 33063 . . . . . . . . . . 11 (((πœ‘ ∧ β„Ž ∈ (SubGrpβ€˜πΊ)) ∧ 𝑁 βŠ† β„Ž) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ 𝑇)
1413anasss 466 . . . . . . . . . 10 ((πœ‘ ∧ (β„Ž ∈ (SubGrpβ€˜πΊ) ∧ 𝑁 βŠ† β„Ž)) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ 𝑇)
1514, 5eleqtrdi 2838 . . . . . . . . 9 ((πœ‘ ∧ (β„Ž ∈ (SubGrpβ€˜πΊ) ∧ 𝑁 βŠ† β„Ž)) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ (SubGrpβ€˜π‘„))
163, 15sylan2b 593 . . . . . . . 8 ((πœ‘ ∧ β„Ž ∈ 𝑆) β†’ ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ∈ (SubGrpβ€˜π‘„))
1716adantr 480 . . . . . . 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 2828 . . . . . 6 (((πœ‘ ∧ β„Ž ∈ 𝑆) ∧ 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))) β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
1918r19.29an 3153 . . . . 5 ((πœ‘ ∧ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))) β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
20 sseq2 4004 . . . . . . . 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 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 ∈ (NrmSGrpβ€˜πΊ))
22 simpr 484 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
234, 8, 9, 21, 22nsgmgclem 33061 . . . . . . . 8 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∈ (SubGrpβ€˜πΊ))
245eleq2i 2820 . . . . . . . . 9 (𝑓 ∈ 𝑇 ↔ 𝑓 ∈ (SubGrpβ€˜π‘„))
25 nsgsubg 19104 . . . . . . . . . . . . 13 (𝑁 ∈ (NrmSGrpβ€˜πΊ) β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
2612, 25syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
274subgss 19073 . . . . . . . . . . . 12 (𝑁 ∈ (SubGrpβ€˜πΊ) β†’ 𝑁 βŠ† 𝐡)
2826, 27syl 17 . . . . . . . . . . 11 (πœ‘ β†’ 𝑁 βŠ† 𝐡)
2928adantr 480 . . . . . . . . . 10 ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ 𝑁 βŠ† 𝐡)
3026ad2antrr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
319grplsmid 33053 . . . . . . . . . . . 12 ((𝑁 ∈ (SubGrpβ€˜πΊ) ∧ π‘Ž ∈ 𝑁) β†’ ({π‘Ž} βŠ• 𝑁) = 𝑁)
3230, 31sylancom 587 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ ({π‘Ž} βŠ• 𝑁) = 𝑁)
3324biimpi 215 . . . . . . . . . . . . 13 (𝑓 ∈ 𝑇 β†’ 𝑓 ∈ (SubGrpβ€˜π‘„))
348nsgqus0 33060 . . . . . . . . . . . . 13 ((𝑁 ∈ (NrmSGrpβ€˜πΊ) ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 ∈ 𝑓)
3512, 33, 34syl2an 595 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ 𝑁 ∈ 𝑓)
3635adantr 480 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ 𝑁 ∈ 𝑓)
3732, 36eqeltrd 2828 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ 𝑇) ∧ π‘Ž ∈ 𝑁) β†’ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓)
3829, 37ssrabdv 4067 . . . . . . . . 9 ((πœ‘ ∧ 𝑓 ∈ 𝑇) β†’ 𝑁 βŠ† {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
3924, 38sylan2br 594 . . . . . . . 8 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑁 βŠ† {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓})
4020, 23, 39elrabd 3682 . . . . . . 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 2839 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ∈ 𝑆)
42 mpteq1 5235 . . . . . . . . 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 5934 . . . . . . . 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 2738 . . . . . . 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 481 . . . . . 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 2727 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘„) = (Baseβ€˜π‘„)
4746subgss 19073 . . . . . . . . . . . . . 14 (𝑓 ∈ (SubGrpβ€˜π‘„) β†’ 𝑓 βŠ† (Baseβ€˜π‘„))
4847adantl 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑓 βŠ† (Baseβ€˜π‘„))
4948sselda 3978 . . . . . . . . . . . 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 7449 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ (𝐺 ~QG 𝑁) ∈ V)
53 subgrcl 19077 . . . . . . . . . . . . . . 15 (𝑁 ∈ (SubGrpβ€˜πΊ) β†’ 𝐺 ∈ Grp)
5426, 53syl 17 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐺 ∈ Grp)
5554ad2antrr 725 . . . . . . . . . . . . 13 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝐺 ∈ Grp)
5650, 51, 52, 55qusbas 17518 . . . . . . . . . . . 12 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ (𝐡 / (𝐺 ~QG 𝑁)) = (Baseβ€˜π‘„))
5749, 56eleqtrrd 2831 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) β†’ 𝑖 ∈ (𝐡 / (𝐺 ~QG 𝑁)))
58 elqsi 8780 . . . . . . . . . . 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 4634 . . . . . . . . . . . . . . . 16 (π‘Ž = π‘₯ β†’ {π‘Ž} = {π‘₯})
6160oveq1d 7429 . . . . . . . . . . . . . . 15 (π‘Ž = π‘₯ β†’ ({π‘Ž} βŠ• 𝑁) = ({π‘₯} βŠ• 𝑁))
6261eleq1d 2813 . . . . . . . . . . . . . 14 (π‘Ž = π‘₯ β†’ (({π‘Ž} βŠ• 𝑁) ∈ 𝑓 ↔ ({π‘₯} βŠ• 𝑁) ∈ 𝑓))
63 simplr 768 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ π‘₯ ∈ 𝐡)
64 simpr 484 . . . . . . . . . . . . . . . 16 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑖 = [π‘₯](𝐺 ~QG 𝑁))
6526ad4antr 731 . . . . . . . . . . . . . . . . 17 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑁 ∈ (SubGrpβ€˜πΊ))
664, 9, 65, 63quslsm 33055 . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑖 = ({π‘₯} βŠ• 𝑁))
68 simpllr 775 . . . . . . . . . . . . . . 15 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ 𝑖 ∈ 𝑓)
6967, 68eqeltrrd 2829 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 ∈ 𝑓) ∧ π‘₯ ∈ 𝐡) ∧ 𝑖 = [π‘₯](𝐺 ~QG 𝑁)) β†’ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)
7062, 63, 69elrabd 3682 . . . . . . . . . . . . 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 511 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 3159 . . . . . . . . . 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 768 . . . . . . . . . . . 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 3680 . . . . . . . . . . . 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 775 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ π‘₯ ∈ 𝐡) ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓) β†’ 𝑖 = ({π‘₯} βŠ• 𝑁))
79 simpr 484 . . . . . . . . . . . . . 14 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ π‘₯ ∈ 𝐡) ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓) β†’ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)
8078, 79eqeltrd 2828 . . . . . . . . . . . . 13 (((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ π‘₯ ∈ 𝐡) ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓) β†’ 𝑖 ∈ 𝑓)
8180anasss 466 . . . . . . . . . . . 12 ((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) ∧ (π‘₯ ∈ 𝐡 ∧ ({π‘₯} βŠ• 𝑁) ∈ 𝑓)) β†’ 𝑖 ∈ 𝑓)
8281adantllr 718 . . . . . . . . . . 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 686 . . . . . . . . . 10 ((((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}) ∧ 𝑖 = ({π‘₯} βŠ• 𝑁)) β†’ 𝑖 ∈ 𝑓)
8483r19.29an 3153 . . . . . . . . 9 (((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) ∧ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁)) β†’ 𝑖 ∈ 𝑓)
8574, 84impbida 800 . . . . . . . 8 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ (𝑖 ∈ 𝑓 ↔ βˆƒπ‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓}𝑖 = ({π‘₯} βŠ• 𝑁)))
86 eqid 2727 . . . . . . . . . 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 5952 . . . . . . . . 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 3475 . . . . . . . 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 289 . . . . . . 7 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ (𝑖 ∈ 𝑓 ↔ 𝑖 ∈ ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁))))
9089eqrdv 2725 . . . . . 6 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ 𝑓 = ran (π‘₯ ∈ {π‘Ž ∈ 𝐡 ∣ ({π‘Ž} βŠ• 𝑁) ∈ 𝑓} ↦ ({π‘₯} βŠ• 𝑁)))
9141, 45, 90rspcedvd 3609 . . . . 5 ((πœ‘ ∧ 𝑓 ∈ (SubGrpβ€˜π‘„)) β†’ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)))
9219, 91impbida 800 . . . 4 (πœ‘ β†’ (βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁)) ↔ 𝑓 ∈ (SubGrpβ€˜π‘„)))
9392abbidv 2796 . . 3 (πœ‘ β†’ {𝑓 ∣ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))} = {𝑓 ∣ 𝑓 ∈ (SubGrpβ€˜π‘„)})
9410rnmpt 5951 . . 3 ran 𝐸 = {𝑓 ∣ βˆƒβ„Ž ∈ 𝑆 𝑓 = ran (π‘₯ ∈ β„Ž ↦ ({π‘₯} βŠ• 𝑁))}
95 abid1 2865 . . 3 (SubGrpβ€˜π‘„) = {𝑓 ∣ 𝑓 ∈ (SubGrpβ€˜π‘„)}
9693, 94, 953eqtr4g 2792 . 2 (πœ‘ β†’ ran 𝐸 = (SubGrpβ€˜π‘„))
9796, 5eqtr4di 2785 1 (πœ‘ β†’ ran 𝐸 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  {cab 2704  βˆƒwrex 3065  {crab 3427  Vcvv 3469   βŠ† wss 3944  {csn 4624   ↦ cmpt 5225  ran crn 5673  β€˜cfv 6542  (class class class)co 7414  [cec 8716   / cqs 8717  Basecbs 17171  lecple 17231   /s cqus 17478  toInccipo 18510  Grpcgrp 18881  SubGrpcsubg 19066  NrmSGrpcnsg 19067   ~QG cqg 19068  LSSumclsm 19580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  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 5143  df-opab 5205  df-mpt 5226  df-tr 5260  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  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 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-sup 9457  df-inf 9458  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-ress 17201  df-plusg 17237  df-mulr 17238  df-sca 17240  df-vsca 17241  df-ip 17242  df-tset 17243  df-ple 17244  df-ds 17246  df-0g 17414  df-imas 17481  df-qus 17482  df-mgm 18591  df-sgrp 18670  df-mnd 18686  df-submnd 18732  df-grp 18884  df-minusg 18885  df-subg 19069  df-nsg 19070  df-eqg 19071  df-oppg 19288  df-lsm 19582
This theorem is referenced by:  nsgqusf1o  33066
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