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Theorem nsgqusf1olem1 32191
Description: Lemma for nsgqusf1o 32194. (Contributed by Thierry Arnoux, 4-Aug-2024.)
Hypotheses
Ref Expression
nsgqusf1o.b 𝐵 = (Base‘𝐺)
nsgqusf1o.s 𝑆 = { ∈ (SubGrp‘𝐺) ∣ 𝑁}
nsgqusf1o.t 𝑇 = (SubGrp‘𝑄)
nsgqusf1o.1 = (le‘(toInc‘𝑆))
nsgqusf1o.2 = (le‘(toInc‘𝑇))
nsgqusf1o.q 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
nsgqusf1o.p = (LSSum‘𝐺)
nsgqusf1o.e 𝐸 = (𝑆 ↦ ran (𝑥 ↦ ({𝑥} 𝑁)))
nsgqusf1o.f 𝐹 = (𝑓𝑇 ↦ {𝑎𝐵 ∣ ({𝑎} 𝑁) ∈ 𝑓})
nsgqusf1o.n (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
Assertion
Ref Expression
nsgqusf1olem1 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
Distinct variable groups:   ,𝑎,𝑓,,𝑥   𝐵,𝑎,𝑓,,𝑥   𝐸,𝑎,𝑓,,𝑥   𝑓,𝐹,,𝑥   𝐺,𝑎,𝑓,,𝑥   𝑁,𝑎,𝑓,,𝑥   𝑄,𝑎,𝑓,,𝑥   𝑆,𝑎,𝑓,,𝑥   𝑇,𝑎,𝑓,,𝑥   𝜑,𝑎,𝑓,,𝑥
Allowed substitution hints:   𝐹(𝑎)   (𝑥,𝑓,,𝑎)   (𝑥,𝑓,,𝑎)

Proof of Theorem nsgqusf1olem1
Dummy variables 𝑖 𝑗 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nsgqusf1o.n . . . . 5 (𝜑𝑁 ∈ (NrmSGrp‘𝐺))
2 nsgqusf1o.q . . . . . 6 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁))
32qusgrp 18985 . . . . 5 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑄 ∈ Grp)
41, 3syl 17 . . . 4 (𝜑𝑄 ∈ Grp)
54ad2antrr 724 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → 𝑄 ∈ Grp)
6 nsgqusf1o.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
76subgss 18929 . . . . . . . . 9 ( ∈ (SubGrp‘𝐺) → 𝐵)
87ad2antlr 725 . . . . . . . 8 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → 𝐵)
98sselda 3944 . . . . . . 7 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → 𝑥𝐵)
10 ovex 7390 . . . . . . . 8 (𝐺 ~QG 𝑁) ∈ V
1110ecelqsi 8712 . . . . . . 7 (𝑥𝐵 → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
129, 11syl 17 . . . . . 6 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → [𝑥](𝐺 ~QG 𝑁) ∈ (𝐵 / (𝐺 ~QG 𝑁)))
13 nsgqusf1o.p . . . . . . 7 = (LSSum‘𝐺)
14 nsgsubg 18960 . . . . . . . . 9 (𝑁 ∈ (NrmSGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺))
151, 14syl 17 . . . . . . . 8 (𝜑𝑁 ∈ (SubGrp‘𝐺))
1615ad3antrrr 728 . . . . . . 7 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → 𝑁 ∈ (SubGrp‘𝐺))
176, 13, 16, 9quslsm 32186 . . . . . 6 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
182a1i 11 . . . . . . . 8 (𝜑𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)))
196a1i 11 . . . . . . . 8 (𝜑𝐵 = (Base‘𝐺))
20 ovexd 7392 . . . . . . . 8 (𝜑 → (𝐺 ~QG 𝑁) ∈ V)
21 subgrcl 18933 . . . . . . . . 9 (𝑁 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
2215, 21syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Grp)
2318, 19, 20, 22qusbas 17427 . . . . . . 7 (𝜑 → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
2423ad3antrrr 728 . . . . . 6 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → (𝐵 / (𝐺 ~QG 𝑁)) = (Base‘𝑄))
2512, 17, 243eltr3d 2852 . . . . 5 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ({𝑥} 𝑁) ∈ (Base‘𝑄))
2625ralrimiva 3143 . . . 4 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ∀𝑥 ({𝑥} 𝑁) ∈ (Base‘𝑄))
27 eqid 2736 . . . . 5 (𝑥 ↦ ({𝑥} 𝑁)) = (𝑥 ↦ ({𝑥} 𝑁))
2827rnmptss 7070 . . . 4 (∀𝑥 ({𝑥} 𝑁) ∈ (Base‘𝑄) → ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄))
2926, 28syl 17 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄))
30 nfv 1917 . . . 4 𝑥((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁)
31 ovexd 7392 . . . 4 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ({𝑥} 𝑁) ∈ V)
32 eqid 2736 . . . . . . 7 (0g𝐺) = (0g𝐺)
3332subg0cl 18936 . . . . . 6 ( ∈ (SubGrp‘𝐺) → (0g𝐺) ∈ )
3433ne0d 4295 . . . . 5 ( ∈ (SubGrp‘𝐺) → ≠ ∅)
3534ad2antlr 725 . . . 4 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ≠ ∅)
3630, 31, 27, 35rnmptn0 6196 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ≠ ∅)
37 nfmpt1 5213 . . . . . . . 8 𝑥(𝑥 ↦ ({𝑥} 𝑁))
3837nfrn 5907 . . . . . . 7 𝑥ran (𝑥 ↦ ({𝑥} 𝑁))
3938nfel2 2925 . . . . . 6 𝑥 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4030, 39nfan 1902 . . . . 5 𝑥(((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
4138nfel2 2925 . . . . . . 7 𝑥(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4238, 41nfralw 3294 . . . . . 6 𝑥𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4338nfel2 2925 . . . . . 6 𝑥((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))
4442, 43nfan 1902 . . . . 5 𝑥(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
45 sneq 4596 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → {𝑥} = {𝑧})
4645oveq1d 7372 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ({𝑥} 𝑁) = ({𝑧} 𝑁))
4746cbvmptv 5218 . . . . . . . . . . 11 (𝑥 ↦ ({𝑥} 𝑁)) = (𝑧 ↦ ({𝑧} 𝑁))
48 simp-4r 782 . . . . . . . . . . . . . 14 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ∈ (SubGrp‘𝐺))
4948ad2antrr 724 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → ∈ (SubGrp‘𝐺))
50 simp-4r 782 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑥)
51 simplr 767 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑦)
52 eqid 2736 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
5352subgcl 18938 . . . . . . . . . . . . 13 (( ∈ (SubGrp‘𝐺) ∧ 𝑥𝑦) → (𝑥(+g𝐺)𝑦) ∈ )
5449, 50, 51, 53syl3anc 1371 . . . . . . . . . . . 12 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑥(+g𝐺)𝑦) ∈ )
55 sneq 4596 . . . . . . . . . . . . . . 15 (𝑧 = (𝑥(+g𝐺)𝑦) → {𝑧} = {(𝑥(+g𝐺)𝑦)})
5655oveq1d 7372 . . . . . . . . . . . . . 14 (𝑧 = (𝑥(+g𝐺)𝑦) → ({𝑧} 𝑁) = ({(𝑥(+g𝐺)𝑦)} 𝑁))
5756eqeq2d 2747 . . . . . . . . . . . . 13 (𝑧 = (𝑥(+g𝐺)𝑦) → ((𝑖(+g𝑄)𝑗) = ({𝑧} 𝑁) ↔ (𝑖(+g𝑄)𝑗) = ({(𝑥(+g𝐺)𝑦)} 𝑁)))
5857adantl 482 . . . . . . . . . . . 12 ((((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) ∧ 𝑧 = (𝑥(+g𝐺)𝑦)) → ((𝑖(+g𝑄)𝑗) = ({𝑧} 𝑁) ↔ (𝑖(+g𝑄)𝑗) = ({(𝑥(+g𝐺)𝑦)} 𝑁)))
59 simpr 485 . . . . . . . . . . . . . . . 16 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖 = ({𝑥} 𝑁))
6017adantr 481 . . . . . . . . . . . . . . . 16 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → [𝑥](𝐺 ~QG 𝑁) = ({𝑥} 𝑁))
6159, 60eqtr4d 2779 . . . . . . . . . . . . . . 15 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
6261ad2antrr 724 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑖 = [𝑥](𝐺 ~QG 𝑁))
63 simpr 485 . . . . . . . . . . . . . . 15 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑗 = ({𝑦} 𝑁))
641ad4antr 730 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
6564ad2antrr 724 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑁 ∈ (NrmSGrp‘𝐺))
6665, 14syl 17 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑁 ∈ (SubGrp‘𝐺))
6749, 7syl 17 . . . . . . . . . . . . . . . . 17 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝐵)
6867, 51sseldd 3945 . . . . . . . . . . . . . . . 16 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑦𝐵)
696, 13, 66, 68quslsm 32186 . . . . . . . . . . . . . . 15 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → [𝑦](𝐺 ~QG 𝑁) = ({𝑦} 𝑁))
7063, 69eqtr4d 2779 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑗 = [𝑦](𝐺 ~QG 𝑁))
7162, 70oveq12d 7375 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) = ([𝑥](𝐺 ~QG 𝑁)(+g𝑄)[𝑦](𝐺 ~QG 𝑁)))
729adantr 481 . . . . . . . . . . . . . . 15 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → 𝑥𝐵)
7372ad2antrr 724 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → 𝑥𝐵)
74 eqid 2736 . . . . . . . . . . . . . . 15 (+g𝑄) = (+g𝑄)
752, 6, 52, 74qusadd 18987 . . . . . . . . . . . . . 14 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝐵𝑦𝐵) → ([𝑥](𝐺 ~QG 𝑁)(+g𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝑁))
7665, 73, 68, 75syl3anc 1371 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → ([𝑥](𝐺 ~QG 𝑁)(+g𝑄)[𝑦](𝐺 ~QG 𝑁)) = [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝑁))
7767, 54sseldd 3945 . . . . . . . . . . . . . 14 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑥(+g𝐺)𝑦) ∈ 𝐵)
786, 13, 66, 77quslsm 32186 . . . . . . . . . . . . 13 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → [(𝑥(+g𝐺)𝑦)](𝐺 ~QG 𝑁) = ({(𝑥(+g𝐺)𝑦)} 𝑁))
7971, 76, 783eqtrd 2780 . . . . . . . . . . . 12 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) = ({(𝑥(+g𝐺)𝑦)} 𝑁))
8054, 58, 79rspcedvd 3583 . . . . . . . . . . 11 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → ∃𝑧 (𝑖(+g𝑄)𝑗) = ({𝑧} 𝑁))
81 ovexd 7392 . . . . . . . . . . 11 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) ∈ V)
8247, 80, 81elrnmptd 5916 . . . . . . . . . 10 (((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
8382adantllr 717 . . . . . . . . 9 ((((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) ∧ 𝑦) ∧ 𝑗 = ({𝑦} 𝑁)) → (𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
84 sneq 4596 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → {𝑥} = {𝑦})
8584oveq1d 7372 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ({𝑥} 𝑁) = ({𝑦} 𝑁))
8685cbvmptv 5218 . . . . . . . . . . . 12 (𝑥 ↦ ({𝑥} 𝑁)) = (𝑦 ↦ ({𝑦} 𝑁))
87 ovex 7390 . . . . . . . . . . . 12 ({𝑦} 𝑁) ∈ V
8886, 87elrnmpti 5915 . . . . . . . . . . 11 (𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ ∃𝑦 𝑗 = ({𝑦} 𝑁))
8988biimpi 215 . . . . . . . . . 10 (𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) → ∃𝑦 𝑗 = ({𝑦} 𝑁))
9089adantl 482 . . . . . . . . 9 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → ∃𝑦 𝑗 = ({𝑦} 𝑁))
9183, 90r19.29a 3159 . . . . . . . 8 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → (𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
9291ralrimiva 3143 . . . . . . 7 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
93 eqid 2736 . . . . . . . . . . 11 (invg𝐺) = (invg𝐺)
9493subginvcl 18937 . . . . . . . . . 10 (( ∈ (SubGrp‘𝐺) ∧ 𝑥) → ((invg𝐺)‘𝑥) ∈ )
9594ad5ant24 759 . . . . . . . . 9 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝐺)‘𝑥) ∈ )
96 simpr 485 . . . . . . . . . . . . 13 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → 𝑦 = ((invg𝐺)‘𝑥))
9796sneqd 4598 . . . . . . . . . . . 12 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → {𝑦} = {((invg𝐺)‘𝑥)})
9897oveq1d 7372 . . . . . . . . . . 11 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → ({𝑦} 𝑁) = ({((invg𝐺)‘𝑥)} 𝑁))
998adantr 481 . . . . . . . . . . . . . 14 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → 𝐵)
10094ad4ant24 752 . . . . . . . . . . . . . 14 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ((invg𝐺)‘𝑥) ∈ )
10199, 100sseldd 3945 . . . . . . . . . . . . 13 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → ((invg𝐺)‘𝑥) ∈ 𝐵)
1026, 13, 16, 101quslsm 32186 . . . . . . . . . . . 12 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) → [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg𝐺)‘𝑥)} 𝑁))
103102ad2antrr 724 . . . . . . . . . . 11 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁) = ({((invg𝐺)‘𝑥)} 𝑁))
10498, 103eqtr4d 2779 . . . . . . . . . 10 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → ({𝑦} 𝑁) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
105104eqeq2d 2747 . . . . . . . . 9 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) ∧ 𝑦 = ((invg𝐺)‘𝑥)) → (((invg𝑄)‘𝑖) = ({𝑦} 𝑁) ↔ ((invg𝑄)‘𝑖) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁)))
10661fveq2d 6846 . . . . . . . . . 10 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) = ((invg𝑄)‘[𝑥](𝐺 ~QG 𝑁)))
107 eqid 2736 . . . . . . . . . . . 12 (invg𝑄) = (invg𝑄)
1082, 6, 93, 107qusinv 18989 . . . . . . . . . . 11 ((𝑁 ∈ (NrmSGrp‘𝐺) ∧ 𝑥𝐵) → ((invg𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
10964, 72, 108syl2anc 584 . . . . . . . . . 10 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘[𝑥](𝐺 ~QG 𝑁)) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
110106, 109eqtrd 2776 . . . . . . . . 9 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) = [((invg𝐺)‘𝑥)](𝐺 ~QG 𝑁))
11195, 105, 110rspcedvd 3583 . . . . . . . 8 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ∃𝑦 ((invg𝑄)‘𝑖) = ({𝑦} 𝑁))
112 fvexd 6857 . . . . . . . 8 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) ∈ V)
11386, 111, 112elrnmptd 5916 . . . . . . 7 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)))
11492, 113jca 512 . . . . . 6 (((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → (∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
115114adantllr 717 . . . . 5 ((((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) ∧ 𝑥) ∧ 𝑖 = ({𝑥} 𝑁)) → (∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
116 ovex 7390 . . . . . . . 8 ({𝑥} 𝑁) ∈ V
11727, 116elrnmpti 5915 . . . . . . 7 (𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ↔ ∃𝑥 𝑖 = ({𝑥} 𝑁))
118117biimpi 215 . . . . . 6 (𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) → ∃𝑥 𝑖 = ({𝑥} 𝑁))
119118adantl 482 . . . . 5 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → ∃𝑥 𝑖 = ({𝑥} 𝑁))
12040, 44, 115, 119r19.29af2 3250 . . . 4 ((((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) ∧ 𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))) → (∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
121120ralrimiva 3143 . . 3 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ∀𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))
122 eqid 2736 . . . . 5 (Base‘𝑄) = (Base‘𝑄)
123122, 74, 107issubg2 18943 . . . 4 (𝑄 ∈ Grp → (ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄) ↔ (ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ↦ ({𝑥} 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))))
124123biimpar 478 . . 3 ((𝑄 ∈ Grp ∧ (ran (𝑥 ↦ ({𝑥} 𝑁)) ⊆ (Base‘𝑄) ∧ ran (𝑥 ↦ ({𝑥} 𝑁)) ≠ ∅ ∧ ∀𝑖 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(∀𝑗 ∈ ran (𝑥 ↦ ({𝑥} 𝑁))(𝑖(+g𝑄)𝑗) ∈ ran (𝑥 ↦ ({𝑥} 𝑁)) ∧ ((invg𝑄)‘𝑖) ∈ ran (𝑥 ↦ ({𝑥} 𝑁))))) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
1255, 29, 36, 121, 124syl13anc 1372 . 2 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ (SubGrp‘𝑄))
126 nsgqusf1o.t . 2 𝑇 = (SubGrp‘𝑄)
127125, 126eleqtrrdi 2849 1 (((𝜑 ∈ (SubGrp‘𝐺)) ∧ 𝑁) → ran (𝑥 ↦ ({𝑥} 𝑁)) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282  {csn 4586  cmpt 5188  ran crn 5634  cfv 6496  (class class class)co 7357  [cec 8646   / cqs 8647  Basecbs 17083  +gcplusg 17133  lecple 17140  0gc0g 17321   /s cqus 17387  toInccipo 18416  Grpcgrp 18748  invgcminusg 18749  SubGrpcsubg 18922  NrmSGrpcnsg 18923   ~QG cqg 18924  LSSumclsm 19416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-ec 8650  df-qs 8654  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-0g 17323  df-imas 17390  df-qus 17391  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-nsg 18926  df-eqg 18927  df-oppg 19124  df-lsm 19418
This theorem is referenced by:  nsgqusf1olem2  32192  nsgqusf1olem3  32193
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