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Mirrors > Home > MPE Home > Th. List > gsummptf1o | Structured version Visualization version GIF version |
Description: Re-index a finite group sum using a bijection. (Contributed by Thierry Arnoux, 29-Mar-2018.) |
Ref | Expression |
---|---|
gsummptf1o.x | ⊢ Ⅎ𝑥𝐻 |
gsummptf1o.b | ⊢ 𝐵 = (Base‘𝐺) |
gsummptf1o.z | ⊢ 0 = (0g‘𝐺) |
gsummptf1o.i | ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻) |
gsummptf1o.g | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
gsummptf1o.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
gsummptf1o.d | ⊢ (𝜑 → 𝐹 ⊆ 𝐵) |
gsummptf1o.f | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) |
gsummptf1o.e | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) |
gsummptf1o.h | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
Ref | Expression |
---|---|
gsummptf1o | ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsummptf1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsummptf1o.z | . . 3 ⊢ 0 = (0g‘𝐺) | |
3 | gsummptf1o.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
4 | gsummptf1o.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
5 | gsummptf1o.d | . . . . . 6 ⊢ (𝜑 → 𝐹 ⊆ 𝐵) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐹 ⊆ 𝐵) |
7 | gsummptf1o.f | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐹) | |
8 | 6, 7 | sseldd 3996 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
9 | 8 | fmpttd 7135 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶):𝐴⟶𝐵) |
10 | eqid 2735 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶) | |
11 | 2 | fvexi 6921 | . . . . 5 ⊢ 0 ∈ V |
12 | 11 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ V) |
13 | 10, 4, 8, 12 | fsuppmptdm 9414 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) finSupp 0 ) |
14 | gsummptf1o.e | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝐸 ∈ 𝐴) | |
15 | 14 | ralrimiva 3144 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴) |
16 | gsummptf1o.h | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) | |
17 | 16 | ralrimiva 3144 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸) |
18 | eqid 2735 | . . . . 5 ⊢ (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸) | |
19 | 18 | f1ompt 7131 | . . . 4 ⊢ ((𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴 ↔ (∀𝑦 ∈ 𝐷 𝐸 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐷 𝑥 = 𝐸)) |
20 | 15, 17, 19 | sylanbrc 583 | . . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸):𝐷–1-1-onto→𝐴) |
21 | 1, 2, 3, 4, 9, 13, 20 | gsumf1o 19949 | . 2 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)))) |
22 | eqidd 2736 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ 𝐸) = (𝑦 ∈ 𝐷 ↦ 𝐸)) | |
23 | eqidd 2736 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
24 | 15, 22, 23 | fmptcos 7151 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶)) |
25 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 ∈ 𝐷) | |
26 | gsummptf1o.x | . . . . . . 7 ⊢ Ⅎ𝑥𝐻 | |
27 | 26 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → Ⅎ𝑥𝐻) |
28 | gsummptf1o.i | . . . . . . 7 ⊢ (𝑥 = 𝐸 → 𝐶 = 𝐻) | |
29 | 28 | adantl 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑥 = 𝐸) → 𝐶 = 𝐻) |
30 | 25, 27, 14, 29 | csbiedf 3939 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ⦋𝐸 / 𝑥⦌𝐶 = 𝐻) |
31 | 30 | mpteq2dva 5248 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐷 ↦ ⦋𝐸 / 𝑥⦌𝐶) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
32 | 24, 31 | eqtrd 2775 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸)) = (𝑦 ∈ 𝐷 ↦ 𝐻)) |
33 | 32 | oveq2d 7447 | . 2 ⊢ (𝜑 → (𝐺 Σg ((𝑥 ∈ 𝐴 ↦ 𝐶) ∘ (𝑦 ∈ 𝐷 ↦ 𝐸))) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
34 | 21, 33 | eqtrd 2775 | 1 ⊢ (𝜑 → (𝐺 Σg (𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝐺 Σg (𝑦 ∈ 𝐷 ↦ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 ∃!wreu 3376 Vcvv 3478 ⦋csb 3908 ⊆ wss 3963 ↦ cmpt 5231 ∘ ccom 5693 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 0gc0g 17486 Σg cgsu 17487 CMndccmn 19813 |
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-rep 5285 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-rmo 3378 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-int 4952 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-se 5642 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-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-card 9977 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-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-cntz 19348 df-cmn 19815 |
This theorem is referenced by: gsummpt2co 33034 gsumhashmul 33047 mdetpmtr1 33784 |
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