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| Mirrors > Home > MPE Home > Th. List > ressmulgnnd | Structured version Visualization version GIF version | ||
| Description: Values for the group multiple function in a restricted structure, a deduction version. (Contributed by metakunt, 14-May-2025.) |
| Ref | Expression |
|---|---|
| ressmulgnnd.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
| ressmulgnnd.2 | ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺)) |
| ressmulgnnd.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
| ressmulgnnd.4 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| Ref | Expression |
|---|---|
| ressmulgnnd | ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ressmulgnnd.4 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 2 | 1 | nngt0d 12315 | . 2 ⊢ (𝜑 → 0 < 𝑁) |
| 3 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑁 ∈ ℕ) |
| 4 | ressmulgnnd.3 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
| 5 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑋 ∈ 𝐴) |
| 6 | ressmulgnnd.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ (Base‘𝐺)) | |
| 7 | eqid 2737 | . . . . . . . . . 10 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴) | |
| 8 | eqid 2737 | . . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 9 | 7, 8 | ressbas2 17283 | . . . . . . . . 9 ⊢ (𝐴 ⊆ (Base‘𝐺) → 𝐴 = (Base‘(𝐺 ↾s 𝐴))) |
| 10 | 6, 9 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 = (Base‘(𝐺 ↾s 𝐴))) |
| 11 | 10 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝐴 = (Base‘(𝐺 ↾s 𝐴))) |
| 12 | ressmulgnnd.1 | . . . . . . . . . 10 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
| 13 | eqcom 2744 | . . . . . . . . . 10 ⊢ (𝐻 = (𝐺 ↾s 𝐴) ↔ (𝐺 ↾s 𝐴) = 𝐻) | |
| 14 | 12, 13 | mpbi 230 | . . . . . . . . 9 ⊢ (𝐺 ↾s 𝐴) = 𝐻 |
| 15 | 14 | fveq2i 6909 | . . . . . . . 8 ⊢ (Base‘(𝐺 ↾s 𝐴)) = (Base‘𝐻) |
| 16 | 15 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝑁) → (Base‘(𝐺 ↾s 𝐴)) = (Base‘𝐻)) |
| 17 | 11, 16 | eqtrd 2777 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝐴 = (Base‘𝐻)) |
| 18 | 5, 17 | eleqtrd 2843 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑋 ∈ (Base‘𝐻)) |
| 19 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
| 20 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
| 21 | eqid 2737 | . . . . . 6 ⊢ (.g‘𝐻) = (.g‘𝐻) | |
| 22 | eqid 2737 | . . . . . 6 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})) | |
| 23 | 19, 20, 21, 22 | mulgnn 19093 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐻)) → (𝑁(.g‘𝐻)𝑋) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁)) |
| 24 | 3, 18, 23 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → (𝑁(.g‘𝐻)𝑋) = (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁)) |
| 25 | fvexd 6921 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐺) ∈ V) | |
| 26 | 25, 6 | ssexd 5324 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
| 27 | eqid 2737 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 28 | 12, 27 | ressplusg 17334 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
| 29 | 26, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) |
| 30 | 29 | eqcomd 2743 | . . . . . . 7 ⊢ (𝜑 → (+g‘𝐻) = (+g‘𝐺)) |
| 31 | 30 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → (+g‘𝐻) = (+g‘𝐺)) |
| 32 | 31 | seqeq2d 14049 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))) |
| 33 | 32 | fveq1d 6908 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
| 34 | 6, 4 | sseldd 3984 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺)) |
| 35 | 34 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑋 ∈ (Base‘𝐺)) |
| 36 | eqid 2737 | . . . . . . 7 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
| 37 | eqid 2737 | . . . . . . 7 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
| 38 | 8, 27, 36, 37 | mulgnn 19093 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑁(.g‘𝐺)𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
| 39 | 3, 35, 38 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → (𝑁(.g‘𝐺)𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
| 40 | 39 | eqcomd 2743 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁) = (𝑁(.g‘𝐺)𝑋)) |
| 41 | 24, 33, 40 | 3eqtrd 2781 | . . 3 ⊢ ((𝜑 ∧ 0 < 𝑁) → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋)) |
| 42 | 41 | ex 412 | . 2 ⊢ (𝜑 → (0 < 𝑁 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋))) |
| 43 | 2, 42 | mpd 15 | 1 ⊢ (𝜑 → (𝑁(.g‘𝐻)𝑋) = (𝑁(.g‘𝐺)𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 {csn 4626 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 < clt 11295 ℕcn 12266 seqcseq 14042 Basecbs 17247 ↾s cress 17274 +gcplusg 17297 .gcmg 19085 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulg 19086 |
| This theorem is referenced by: 2sqr3minply 33791 aks6d1c6lem4 42174 unitscyglem5 42200 |
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