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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 12313 | . 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 2735 | . . . . . . . . . 10 ⊢ (𝐺 ↾s 𝐴) = (𝐺 ↾s 𝐴) | |
8 | eqid 2735 | . . . . . . . . . 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 2742 | . . . . . . . . . 10 ⊢ (𝐻 = (𝐺 ↾s 𝐴) ↔ (𝐺 ↾s 𝐴) = 𝐻) | |
14 | 12, 13 | mpbi 230 | . . . . . . . . 9 ⊢ (𝐺 ↾s 𝐴) = 𝐻 |
15 | 14 | fveq2i 6910 | . . . . . . . 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 2775 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝐴 = (Base‘𝐻)) |
18 | 5, 17 | eleqtrd 2841 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑋 ∈ (Base‘𝐻)) |
19 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
20 | eqid 2735 | . . . . . 6 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
21 | eqid 2735 | . . . . . 6 ⊢ (.g‘𝐻) = (.g‘𝐻) | |
22 | eqid 2735 | . . . . . 6 ⊢ seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})) | |
23 | 19, 20, 21, 22 | mulgnn 19106 | . . . . 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 6922 | . . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝐺) ∈ V) | |
26 | 25, 6 | ssexd 5330 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ V) |
27 | eqid 2735 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
28 | 12, 27 | ressplusg 17336 | . . . . . . . . 9 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
29 | 26, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) |
30 | 29 | eqcomd 2741 | . . . . . . 7 ⊢ (𝜑 → (+g‘𝐻) = (+g‘𝐺)) |
31 | 30 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → (+g‘𝐻) = (+g‘𝐺)) |
32 | 31 | seqeq2d 14046 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → seq1((+g‘𝐻), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋}))) |
33 | 32 | fveq1d 6909 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → (seq1((+g‘𝐻), (ℕ × {𝑋}))‘𝑁) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
34 | 6, 4 | sseldd 3996 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐺)) |
35 | 34 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑋 ∈ (Base‘𝐺)) |
36 | eqid 2735 | . . . . . . 7 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
37 | eqid 2735 | . . . . . . 7 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
38 | 8, 27, 36, 37 | mulgnn 19106 | . . . . . 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 2741 | . . . 4 ⊢ ((𝜑 ∧ 0 < 𝑁) → (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁) = (𝑁(.g‘𝐺)𝑋)) |
41 | 24, 33, 40 | 3eqtrd 2779 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 {csn 4631 class class class wbr 5148 × cxp 5687 ‘cfv 6563 (class class class)co 7431 0cc0 11153 1c1 11154 < clt 11293 ℕcn 12264 seqcseq 14039 Basecbs 17245 ↾s cress 17274 +gcplusg 17298 .gcmg 19098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulg 19099 |
This theorem is referenced by: 2sqr3minply 33753 aks6d1c6lem4 42155 unitscyglem5 42181 |
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