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Mirrors > Home > MPE Home > Th. List > Mathboxes > int-eqmvtd | Structured version Visualization version GIF version |
Description: EquMoveTerm generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.) |
Ref | Expression |
---|---|
int-eqmvtd.1 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
int-eqmvtd.2 | ⊢ (𝜑 → 𝐷 ∈ ℝ) |
int-eqmvtd.3 | ⊢ (𝜑 → 𝐴 = 𝐵) |
int-eqmvtd.4 | ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷)) |
Ref | Expression |
---|---|
int-eqmvtd | ⊢ (𝜑 → 𝐶 = (𝐵 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | int-eqmvtd.3 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | int-eqmvtd.4 | . . . . 5 ⊢ (𝜑 → 𝐴 = (𝐶 + 𝐷)) | |
3 | 1, 2 | eqtr3d 2781 | . . . 4 ⊢ (𝜑 → 𝐵 = (𝐶 + 𝐷)) |
4 | 3 | oveq1d 7250 | . . 3 ⊢ (𝜑 → (𝐵 − 𝐷) = ((𝐶 + 𝐷) − 𝐷)) |
5 | int-eqmvtd.1 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | 5 | recnd 10891 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
7 | int-eqmvtd.2 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ) | |
8 | 7 | recnd 10891 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
9 | 6, 8 | pncand 11220 | . . 3 ⊢ (𝜑 → ((𝐶 + 𝐷) − 𝐷) = 𝐶) |
10 | 4, 9 | eqtrd 2779 | . 2 ⊢ (𝜑 → (𝐵 − 𝐷) = 𝐶) |
11 | 10 | eqcomd 2745 | 1 ⊢ (𝜑 → 𝐶 = (𝐵 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 (class class class)co 7235 ℝcr 10758 + caddc 10762 − cmin 11092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 ax-un 7545 ax-resscn 10816 ax-1cn 10817 ax-icn 10818 ax-addcl 10819 ax-addrcl 10820 ax-mulcl 10821 ax-mulrcl 10822 ax-mulcom 10823 ax-addass 10824 ax-mulass 10825 ax-distr 10826 ax-i2m1 10827 ax-1ne0 10828 ax-1rid 10829 ax-rnegex 10830 ax-rrecex 10831 ax-cnre 10832 ax-pre-lttri 10833 ax-pre-lttrn 10834 ax-pre-ltadd 10835 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-po 5486 df-so 5487 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-riota 7192 df-ov 7238 df-oprab 7239 df-mpo 7240 df-er 8415 df-en 8651 df-dom 8652 df-sdom 8653 df-pnf 10899 df-mnf 10900 df-ltxr 10902 df-sub 11094 |
This theorem is referenced by: (None) |
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