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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubidaddid1lem | Structured version Visualization version GIF version |
Description: Lemma for resubidaddid1 39301. A special case of npncan 10900. (Contributed by Steven Nguyen, 8-Jan-2023.) |
Ref | Expression |
---|---|
resubidaddid1lem.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resubidaddid1lem.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
resubidaddid1lem.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
resubidaddid1lem.1 | ⊢ (𝜑 → (𝐴 −ℝ 𝐵) = (𝐵 −ℝ 𝐶)) |
Ref | Expression |
---|---|
resubidaddid1lem | ⊢ (𝜑 → ((𝐴 −ℝ 𝐵) + (𝐵 −ℝ 𝐶)) = (𝐴 −ℝ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubidaddid1lem.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
2 | resubidaddid1lem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | resubidaddid1lem.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | rersubcl 39284 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) ∈ ℝ) | |
5 | 2, 3, 4 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐴 −ℝ 𝐵) ∈ ℝ) |
6 | rersubcl 39284 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐵 −ℝ 𝐶) ∈ ℝ) | |
7 | 3, 1, 6 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐵 −ℝ 𝐶) ∈ ℝ) |
8 | 5, 7 | readdcld 10663 | . 2 ⊢ (𝜑 → ((𝐴 −ℝ 𝐵) + (𝐵 −ℝ 𝐶)) ∈ ℝ) |
9 | resubidaddid1lem.1 | . . . . . 6 ⊢ (𝜑 → (𝐴 −ℝ 𝐵) = (𝐵 −ℝ 𝐶)) | |
10 | 9 | eqcomd 2826 | . . . . 5 ⊢ (𝜑 → (𝐵 −ℝ 𝐶) = (𝐴 −ℝ 𝐵)) |
11 | 3, 1, 5 | resubaddd 39286 | . . . . 5 ⊢ (𝜑 → ((𝐵 −ℝ 𝐶) = (𝐴 −ℝ 𝐵) ↔ (𝐶 + (𝐴 −ℝ 𝐵)) = 𝐵)) |
12 | 10, 11 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝐶 + (𝐴 −ℝ 𝐵)) = 𝐵) |
13 | 12 | oveq1d 7164 | . . 3 ⊢ (𝜑 → ((𝐶 + (𝐴 −ℝ 𝐵)) + (𝐵 −ℝ 𝐶)) = (𝐵 + (𝐵 −ℝ 𝐶))) |
14 | 1 | recnd 10662 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
15 | 5 | recnd 10662 | . . . 4 ⊢ (𝜑 → (𝐴 −ℝ 𝐵) ∈ ℂ) |
16 | 7 | recnd 10662 | . . . 4 ⊢ (𝜑 → (𝐵 −ℝ 𝐶) ∈ ℂ) |
17 | 14, 15, 16 | addassd 10656 | . . 3 ⊢ (𝜑 → ((𝐶 + (𝐴 −ℝ 𝐵)) + (𝐵 −ℝ 𝐶)) = (𝐶 + ((𝐴 −ℝ 𝐵) + (𝐵 −ℝ 𝐶)))) |
18 | 2, 3, 7 | resubaddd 39286 | . . . 4 ⊢ (𝜑 → ((𝐴 −ℝ 𝐵) = (𝐵 −ℝ 𝐶) ↔ (𝐵 + (𝐵 −ℝ 𝐶)) = 𝐴)) |
19 | 9, 18 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐵 + (𝐵 −ℝ 𝐶)) = 𝐴) |
20 | 13, 17, 19 | 3eqtr3d 2863 | . 2 ⊢ (𝜑 → (𝐶 + ((𝐴 −ℝ 𝐵) + (𝐵 −ℝ 𝐶))) = 𝐴) |
21 | 1, 8, 20 | reladdrsub 39291 | 1 ⊢ (𝜑 → ((𝐴 −ℝ 𝐵) + (𝐵 −ℝ 𝐶)) = (𝐴 −ℝ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 (class class class)co 7149 ℝcr 10529 + caddc 10533 −ℝ cresub 39271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-addrcl 10591 ax-addass 10595 ax-rnegex 10601 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 df-resub 39272 |
This theorem is referenced by: (None) |
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