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Mirrors > Home > ILE Home > Th. List > dvmptsubcn | GIF version |
Description: Function-builder for derivative, subtraction rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 31-Dec-2023.) |
Ref | Expression |
---|---|
dvmptcmulcn.a | ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) |
dvmptcmulcn.b | ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉) |
dvmptcmulcn.da | ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵)) |
dvmptsubcn.c | ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ) |
dvmptsubcn.d | ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐷 ∈ 𝑊) |
dvmptsubcn.dc | ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐶)) = (𝑥 ∈ ℂ ↦ 𝐷)) |
Ref | Expression |
---|---|
dvmptsubcn | ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵 − 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnelprrecn 7903 | . . . 4 ⊢ ℂ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ}) |
3 | dvmptcmulcn.a | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐴 ∈ ℂ) | |
4 | dvmptcmulcn.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ 𝑉) | |
5 | dvmptcmulcn.da | . . 3 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐴)) = (𝑥 ∈ ℂ ↦ 𝐵)) | |
6 | ssidd 3168 | . . 3 ⊢ (𝜑 → ℂ ⊆ ℂ) | |
7 | dvmptsubcn.c | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ) | |
8 | 7 | negcld 8210 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → -𝐶 ∈ ℂ) |
9 | dvmptsubcn.d | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐷 ∈ 𝑊) | |
10 | dvmptsubcn.dc | . . . . 5 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ 𝐶)) = (𝑥 ∈ ℂ ↦ 𝐷)) | |
11 | 2, 7, 9, 10, 6 | dvmptclx 13439 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐷 ∈ ℂ) |
12 | 11 | negcld 8210 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → -𝐷 ∈ ℂ) |
13 | 7, 9, 10 | dvmptnegcn 13443 | . . 3 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ -𝐶)) = (𝑥 ∈ ℂ ↦ -𝐷)) |
14 | 2, 3, 4, 5, 6, 8, 12, 13 | dvmptaddx 13440 | . 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 + -𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵 + -𝐷))) |
15 | 3, 7 | negsubd 8229 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐴 + -𝐶) = (𝐴 − 𝐶)) |
16 | 15 | mpteq2dva 4077 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝐴 + -𝐶)) = (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶))) |
17 | 16 | oveq2d 5867 | . 2 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 + -𝐶))) = (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶)))) |
18 | 2, 3, 4, 5, 6 | dvmptclx 13439 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) |
19 | 18, 11 | negsubd 8229 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝐵 + -𝐷) = (𝐵 − 𝐷)) |
20 | 19 | mpteq2dva 4077 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝐵 + -𝐷)) = (𝑥 ∈ ℂ ↦ (𝐵 − 𝐷))) |
21 | 14, 17, 20 | 3eqtr3d 2211 | 1 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝐴 − 𝐶))) = (𝑥 ∈ ℂ ↦ (𝐵 − 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 {cpr 3582 ↦ cmpt 4048 (class class class)co 5851 ℂcc 7765 ℝcr 7766 + caddc 7770 − cmin 8083 -cneg 8084 D cdv 13383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 ax-arch 7886 ax-caucvg 7887 ax-addf 7889 ax-mulf 7890 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-isom 5205 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-of 6059 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-map 6626 df-pm 6627 df-sup 6959 df-inf 6960 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-n0 9129 df-z 9206 df-uz 9481 df-q 9572 df-rp 9604 df-xneg 9722 df-xadd 9723 df-seqfrec 10395 df-exp 10469 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 df-rest 12574 df-topgen 12593 df-psmet 12746 df-xmet 12747 df-met 12748 df-bl 12749 df-mopn 12750 df-top 12755 df-topon 12768 df-bases 12800 df-ntr 12855 df-cn 12947 df-cnp 12948 df-tx 13012 df-cncf 13317 df-limced 13384 df-dvap 13385 |
This theorem is referenced by: dvef 13447 |
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