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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-00id | Structured version Visualization version GIF version |
Description: 00id 10808 proven without ax-mulcom 10594 but using ax-1ne0 10599. (Though note that the current version of 00id 10808 can be changed to avoid ax-icn 10589, ax-addcl 10590, ax-mulcl 10592, ax-i2m1 10598, ax-cnre 10603. Most of this is by using 0cnALT3 39229 instead of 0cn 10626). (Contributed by SN, 25-Dec-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
sn-00id | ⊢ (0 + 0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10636 | . . . 4 ⊢ 0 ∈ ℝ | |
2 | resubadd 39285 | . . . 4 ⊢ ((0 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 −ℝ 0) = 0 ↔ (0 + 0) = 0)) | |
3 | 1, 1, 1, 2 | mp3an 1456 | . . 3 ⊢ ((0 −ℝ 0) = 0 ↔ (0 + 0) = 0) |
4 | df-ne 3016 | . . . 4 ⊢ ((0 −ℝ 0) ≠ 0 ↔ ¬ (0 −ℝ 0) = 0) | |
5 | sn-00idlem2 39305 | . . . . 5 ⊢ ((0 −ℝ 0) ≠ 0 → (0 −ℝ 0) = 1) | |
6 | sn-00idlem3 39306 | . . . . 5 ⊢ ((0 −ℝ 0) = 1 → (0 + 0) = 0) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((0 −ℝ 0) ≠ 0 → (0 + 0) = 0) |
8 | 4, 7 | sylbir 237 | . . 3 ⊢ (¬ (0 −ℝ 0) = 0 → (0 + 0) = 0) |
9 | 3, 8 | sylnbir 333 | . 2 ⊢ (¬ (0 + 0) = 0 → (0 + 0) = 0) |
10 | 9 | pm2.18i 131 | 1 ⊢ (0 + 0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 (class class class)co 7149 ℝcr 10529 0cc0 10530 1c1 10531 + 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-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 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: re0m0e0 39308 |
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