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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1t1e1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of 1t1e1 11989 using a different set of axioms (add ax-mulrcl 10789, ax-i2m1 10794, ax-1ne0 10795, ax-rrecex 10798 and remove ax-resscn 10783, ax-mulcom 10790, ax-mulass 10792, ax-distr 10793). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1t1e1ALT | ⊢ (1 · 1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10830 | . 2 ⊢ 1 ∈ ℝ | |
2 | ax-1rid 10796 | . 2 ⊢ (1 ∈ ℝ → (1 · 1) = 1) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (1 · 1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7210 ℝcr 10725 1c1 10727 · cmul 10731 |
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 2112 ax-9 2120 ax-ext 2708 ax-1cn 10784 ax-icn 10785 ax-addcl 10786 ax-mulcl 10788 ax-mulrcl 10789 ax-i2m1 10794 ax-1ne0 10795 ax-1rid 10796 ax-rrecex 10798 ax-cnre 10799 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2940 df-ral 3063 df-rex 3064 df-rab 3067 df-v 3407 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-br 5051 df-iota 6335 df-fv 6385 df-ov 7213 |
This theorem is referenced by: nnmul1com 40006 remulinvcom 40120 sn-0tie0 40127 |
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