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| Mirrors > Home > ILE Home > Th. List > qusmulval | GIF version | ||
| Description: The multiplication 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) |
| qusmulf.p | ⊢ · = (.r‘𝑅) |
| qusmulf.a | ⊢ ∙ = (.r‘𝑈) |
| Ref | Expression |
|---|---|
| qusmulval | ⊢ ((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 2231 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) | |
| 8 | basfn 13143 | . . . . . . 7 ⊢ Base Fn V | |
| 9 | 4 | elexd 2816 | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ V) |
| 10 | funfvex 5656 | . . . . . . . 8 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V) | |
| 11 | 10 | funfni 5432 | . . . . . . 7 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V) |
| 12 | 8, 9, 11 | sylancr 414 | . . . . . 6 ⊢ (𝜑 → (Base‘𝑅) ∈ V) |
| 13 | 2, 12 | eqeltrd 2308 | . . . . 5 ⊢ (𝜑 → 𝑉 ∈ V) |
| 14 | erex 6726 | . . . . 5 ⊢ ( ∼ Er 𝑉 → (𝑉 ∈ V → ∼ ∈ V)) | |
| 15 | 3, 13, 14 | sylc 62 | . . . 4 ⊢ (𝜑 → ∼ ∈ V) |
| 16 | 1, 2, 7, 15, 4 | qusval 13408 | . . 3 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅)) |
| 17 | 1, 2, 7, 15, 4 | quslem 13409 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ):𝑉–onto→(𝑉 / ∼ )) |
| 18 | qusmulf.p | . . 3 ⊢ · = (.r‘𝑅) | |
| 19 | qusmulf.a | . . 3 ⊢ ∙ = (.r‘𝑈) | |
| 20 | 16, 2, 17, 4, 18, 19 | imasmulr 13394 | . 2 ⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )‘𝑝), ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )‘𝑞)〉, ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )‘(𝑝 · 𝑞))〉}) |
| 21 | mulrslid 13217 | . . . . 5 ⊢ (.r = Slot (.r‘ndx) ∧ (.r‘ndx) ∈ ℕ) | |
| 22 | 21 | slotex 13111 | . . . 4 ⊢ (𝑅 ∈ 𝑍 → (.r‘𝑅) ∈ V) |
| 23 | 4, 22 | syl 14 | . . 3 ⊢ (𝜑 → (.r‘𝑅) ∈ V) |
| 24 | 18, 23 | eqeltrid 2318 | . 2 ⊢ (𝜑 → · ∈ V) |
| 25 | 1, 2, 3, 4, 5, 6, 7, 20, 24 | qusaddvallemg 13418 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ([𝑋] ∼ ∙ [𝑌] ∼ ) = [(𝑋 · 𝑌)] ∼ ) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 Vcvv 2802 class class class wbr 4088 ↦ cmpt 4150 Fn wfn 5321 ‘cfv 5326 (class class class)co 6018 Er wer 6699 [cec 6700 / cqs 6701 Basecbs 13084 .rcmulr 13163 /s cqus 13385 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-addcom 8132 ax-addass 8134 ax-i2m1 8137 ax-0lt1 8138 ax-0id 8140 ax-rnegex 8141 ax-pre-ltirr 8144 ax-pre-lttrn 8146 ax-pre-ltadd 8148 |
| This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-tp 3677 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-er 6702 df-ec 6704 df-qs 6708 df-pnf 8216 df-mnf 8217 df-ltxr 8219 df-inn 9144 df-2 9202 df-3 9203 df-ndx 13087 df-slot 13088 df-base 13090 df-plusg 13175 df-mulr 13176 df-iimas 13387 df-qus 13388 |
| This theorem is referenced by: qusrhm 14545 qusmul2 14546 qusmulrng 14549 |
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