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Mirrors > Home > MPE Home > Th. List > qusaddval | Structured version Visualization version GIF version |
Description: The addition in a quotient structure. (Contributed by Mario Carneiro, 24-Feb-2015.) |
Ref | Expression |
---|---|
qusaddf.u | ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) |
qusaddf.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
qusaddf.r | ⊢ (𝜑 → ∼ Er 𝑉) |
qusaddf.z | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
qusaddf.e | ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) |
qusaddf.c | ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) |
qusaddf.p | ⊢ · = (+g‘𝑅) |
qusaddf.a | ⊢ ∙ = (+g‘𝑈) |
Ref | Expression |
---|---|
qusaddval | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusaddf.u | . 2 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ )) | |
2 | qusaddf.v | . 2 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | qusaddf.r | . 2 ⊢ (𝜑 → ∼ Er 𝑉) | |
4 | qusaddf.z | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
5 | qusaddf.e | . 2 ⊢ (𝜑 → ((𝑎 ∼ 𝑝 ∧ 𝑏 ∼ 𝑞) → (𝑎 · 𝑏) ∼ (𝑝 · 𝑞))) | |
6 | qusaddf.c | . 2 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (𝑝 · 𝑞) ∈ 𝑉) | |
7 | eqid 2731 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
8 | fvex 6904 | . . . . . 6 ⊢ (Base‘𝑅) ∈ V | |
9 | 2, 8 | eqeltrdi 2840 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ V) |
10 | erex 8733 | . . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V)) | |
11 | 3, 9, 10 | sylc 65 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
12 | 1, 2, 7, 11, 4 | qusval 17495 | . . 3 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅)) |
13 | 1, 2, 7, 11, 4 | quslem 17496 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ):𝑉–onto→(𝑉 / ∼ )) |
14 | qusaddf.p | . . 3 ⊢ · = (+g‘𝑅) | |
15 | qusaddf.a | . . 3 ⊢ ∙ = (+g‘𝑈) | |
16 | 12, 2, 13, 4, 14, 15 | imasplusg 17470 | . 2 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )‘𝑝), ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )‘𝑞)〉, ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )‘(𝑝 · 𝑞))〉}) |
17 | 1, 2, 3, 4, 5, 6, 7, 16 | qusaddvallem 17504 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 Vcvv 3473 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 Er wer 8706 [cec 8707 / cqs 8708 Basecbs 17151 +gcplusg 17204 /s cqus 17458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-ec 8711 df-qs 8715 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-imas 17461 df-qus 17462 |
This theorem is referenced by: qusadd 19110 qus0subgadd 19121 frgpadd 19679 pi1addval 24896 |
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